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3.E: Practice

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    Conceptual Questions

    Position, Displacement, and Average Velocity

    1. Give an example in which there are clear distinctions among distance traveled, displacement, and magnitude of displacement. Identify each quantity in your example specifically.
    2. Under what circumstances does distance traveled equal magnitude of displacement? What is the only case in which magnitude of displacement and displacement are exactly the same?
    3. Bacteria move back and forth using their flagella (structures that look like little tails). Speeds of up to 50 \(\mu\)m/s (50 x 10−6 m/s) have been observed. The total distance traveled by a bacterium is large for its size, whereas its displacement is small. Why is this?
    4. Give an example of a device used to measure time and identify what change in that device indicates a change in time.
    5. Does a car’s odometer measure distance traveled or displacement?
    6. During a given time interval the average velocity of an object is zero. What can you say conclude about its displacement over the time interval?

    Instantaneous Velocity and Speed

    1. There is a distinction between average speed and the magnitude of average velocity. Give an example that illustrates the difference between these two quantities.
    2. Does the speedometer of a car measure speed or velocity?
    3. If you divide the total distance traveled on a car trip (as determined by the odometer) by the elapsed time of the trip, are you calculating average speed or magnitude of average velocity? Under what circumstances are these two quantities the same?
    4. How are instantaneous velocity and instantaneous speed related to one another? How do they differ?

    Average and Instantaneous Acceleration

    1. Is it possible for speed to be constant while acceleration is not zero?
    2. Is it possible for velocity to be constant while acceleration is not zero? Explain.
    3. Give an example in which velocity is zero yet acceleration is not.
    4. If a subway train is moving to the left (has a negative velocity) and then comes to a stop, what is the direction of its acceleration? Is the acceleration positive or negative?
    5. Plus and minus signs are used in one-dimensional motion to indicate direction. What is the sign of an acceleration that reduces the magnitude of a negative velocity? Of a positive velocity?

    Displacement and Velocity Vectors

    1. What form does the trajectory of a particle have if the distance from any point A to point B is equal to the magnitude of the displacement from A to B?
    2. Give an example of a trajectory in two or three dimensions caused by independent perpendicular motions.
    3. If the instantaneous velocity is zero, what can be said about the slope of the position function?

    Acceleration Vector

    1. If the position function of a particle is a linear function of time, what can be said about its acceleration?
    2. If an object has a constant x-component of the velocity and suddenly experiences an acceleration in the y direction, does the x-component of its velocity change?
    3. If an object has a constant x-component of velocity and suddenly experiences an acceleration at an angle of 70° in the x direction, does the x-component of velocity change?

    Problems

    Position, Displacement, and Average Velocity

    1. Consider a coordinate system in which the positive x axis is directed upward vertically. What are the positions of a particle (a) 5.0 m directly above the origin and (b) 2.0 m below the origin?
    2. A car is 2.0 km west of a traffic light at t = 0 and 5.0 km east of the light at t = 6.0 min. Assume the origin of the coordinate system is the light and the positive x direction is eastward. (a) What are the car’s position vectors at these two times? (b) What is the car’s displacement between 0 min and 6.0 min?
    3. The Shanghai maglev train connects Longyang Road to Pudong International Airport, a distance of 30 km. The journey takes 8 minutes on average. What is the maglev train’s average velocity?
    4. The position of a particle moving along the x-axis is given by x(t) = 4.0 − 2.0t m. (a) At what time does the particle cross the origin? (b) What is the displacement of the particle between t = 3.0 s and t = 6.0 s?
    5. A cyclist rides 8.0 km east for 20 minutes, then he turns and heads west for 8 minutes and 3.2 km. Finally, he rides east for 16 km, which takes 40 minutes. (a) What is the final displacement of the cyclist? (b) What is his average velocity?
    6. On February 15, 2013, a superbolide meteor (brighter than the Sun) entered Earth’s atmosphere over Chelyabinsk, Russia, and exploded at an altitude of 23.5 km. Eyewitnesses could feel the intense heat from the fireball, and the blast wave from the explosion blew out windows in buildings. The blast wave took approximately 2 minutes 30 seconds to reach ground level. (a) What was the average velocity of the blast wave? b) Compare this with the speed of sound, which is 343 m/s at sea level.

    Instantaneous Velocity and Speed

    1. A woodchuck runs 20 m to the right in 5 s, then turns and runs 10 m to the left in 3 s. (a) What is the average velocity of the woodchuck? (b) What is its average speed?
    2. Sketch the velocity-versus-time graph from the following position-versus-time graph.

    Graph shows position in meters plotted versus time in seconds. It starts at the origin, reaches 4 meters at 0.4 seconds; decreases to -2 meters at 0.6 seconds, reaches minimum of -6 meters at 1 second, increases to -4 meters at 1.6 seconds, and reaches 2 meters at 2 seconds.

    1. Sketch the velocity-versus-time graph from the following position-versus-time graph.

    Graph shows position plotted versus time in seconds. Graph has a sinusoidal shape. It starts with the positive value at zero time, changes to negative, and then starts to increase.

    1. Given the following velocity-versus-time graph, sketch the position-versus-time graph.

    Graph shows velocity plotted versus time. It starts with the positive value at zero time, decreases to the negative value and remains constant.

    1. An object has a position function x(t) = 5t m. (a) What is the velocity as a function of time? (b) Graph the position function and the velocity function. 35. A particle moves along the x-axis according to x(t) = 10t − 2t2 m. (a) What is the instantaneous velocity at t = 2 s and t = 3 s? (b) What is the instantaneous speed at these times? (c) What is the average velocity between t = 2 s and t = 3 s?
    2. Unreasonable results. A particle moves along the x-axis according to x(t) = 3t3 + 5t . At what time is the velocity of the particle equal to zero? Is this reasonable?

    Average and Instantaneous Acceleration

    1. A cheetah can accelerate from rest to a speed of 30.0 m/s in 7.00 s. What is its acceleration?
    2. Dr. John Paul Stapp was a U.S. Air Force officer who studied the effects of extreme acceleration on the human body. On December 10, 1954, Stapp rode a rocket sled, accelerating from rest to a top speed of 282 m/s (1015 km/h) in 5.00 s and was brought jarringly back to rest in only 1.40 s. Calculate his (a) acceleration in his direction of motion and (b) acceleration opposite to his direction of motion. Express each in multiples of g (9.80 m/s2) by taking its ratio to the acceleration of gravity.
    3. Sketch the acceleration-versus-time graph from the following velocity-versus-time graph.

    Graph shows velocity in meters per second plotted versus time in seconds. Velocity is zero and zero seconds, increases to 6 meters per second at 20 seconds, decreases to 2 meters per second at 50 and remains constant until 70 seconds, increases to 4 meters per second at 90 seconds, and decreases to –2 meters per second at 100 seconds.

    1. A commuter backs her car out of her garage with an acceleration of 1.40 m/s2. (a) How long does it take her to reach a speed of 2.00 m/s? (b) If she then brakes to a stop in 0.800 s, what is her acceleration?
    2. Assume an intercontinental ballistic missile goes from rest to a suborbital speed of 6.50 km/s in 60.0 s (the actual speed and time are classified). What is its average acceleration in meters per second and in multiples of g (9.80 m/s2)?
    3. An airplane, starting from rest, moves down the runway at constant acceleration for 18 s and then takes off at a speed of 60 m/s. What is the average acceleration of the plane?

    Displacement and Velocity Vectors

    1. The coordinates of a particle in a rectangular coordinate system are (1.0, –4.0, 6.0). What is the position vector of the particle?
    2. The position of a particle changes from \(\vec{r}_{1}\) = (2.0 \(\hat{i}\) + 3.0 \(\hat{j}\))cm to \(\vec{r}_{2}\) = (−4.0 \(\hat{i}\) + 3.0 \(\hat{j}\)) cm. What is the particle’s displacement?
    3. The 18th hole at Pebble Beach Golf Course is a dogleg to the left of length 496.0 m. The fairway off the tee is taken to be the x direction. A golfer hits his tee shot a distance of 300.0 m, corresponding to a displacement \(\Delta \vec{r}_{1}\) = 300.0 m \(\hat{i}\), and hits his second shot 189.0 m with a displacement \(\Delta \vec{r}_{2}\) = 172.0 m \(\hat{i}\) + 80.3 m \(\hat{j}\). What is the final displacement of the golf ball from the tee?
    4. A bird flies straight northeast a distance of 95.0 km for 3.0 h. With the x-axis due east and the y-axis due north, what is the displacement in unit vector notation for the bird? What is the average velocity for the trip?
    5. A cyclist rides 5.0 km due east, then 10.0 km 20° west of north. From this point she rides 8.0 km due west. What is the final displacement from where the cyclist started?
    6. New York Rangers defenseman Daniel Girardi stands at the goal and passes a hockey puck 20 m and 45° from straight down the ice to left wing Chris Kreider waiting at the blue line. Kreider waits for Girardi to reach the blue line and passes the puck directly across the ice to him 10 m away. What is the final displacement of the puck? See the following figure.

    An illustration of the situation described in the problem. The goal and the two ice hockey players are drawn as viewed from above. The goal and Girardi are at the origin of an x y coordinate system. A gray arrow representing 20 meters at 45 degrees from the positive x direction is shown, with Kreider drawn near the tip of the arrow. A blue line, parallel to the x axis, is also drawn at the tip of this arrow. A second gray arrow is shown starting at the Kreider’s location, pointing horizontally to the left, and representing a distance of 10 meters. A dark blue arrow is drawn from the goal at the origin to the tip of the second, 10 meter, gray arrow.

    1. The position of a particle is \(\vec{r}\)(t) = 4.0t2 \(\hat{i}\) − 3.0 \(\hat{j}\) + 2.0t3 \(\hat{k}\) m. (a) What is the velocity of the particle at 0 s and at 1.0 s? (b) What is the average velocity between 0 s and 1.0 s?
    2. Clay Matthews, a linebacker for the Green Bay Packers, can reach a speed of 10.0 m/s. At the start of a play, Matthews runs downfield at 45° with respect to the 50-yard line and covers 8.0 m in 1 s. He then runs straight down the field at 90° with respect to the 50-yard line for 12 m, with an elapsed time of 1.2 s. (a) What is Matthews’ final displacement from the start of the play? (b) What is his average velocity?
    3. The F-35B Lighting II is a short-takeoff and vertical landing fighter jet. If it does a vertical takeoff to 20.00-m height above the ground and then follows a flight path angled at 30° with respect to the ground for 20.00 km, what is the final displacement?

    Acceleration Vector

    1. The position of a particle is \(\vec{r}\) (t) = (3.0 2 \(\hat{i}\) + 5.0 \(\hat{j}\) − 6.0t \(\hat{k}\)) m. (a) Determine its velocity and acceleration as functions of time. (b) What are its velocity and acceleration at time t = 0?
    2. A particle’s acceleration is (4.0 \(\hat{i}\) + 3.0 \(\hat{j}\))m/s2. At t = 0, its position and velocity are zero. (a) What are the particle’s position and velocity as functions of time? (b) Find the equation of the path of the particle. Draw the x- and y-axes and sketch the trajectory of the particle.
    3. A boat leaves the dock at t = 0 and heads out into a lake with an acceleration of 2.0 m/s2 \(\hat{i}\). A strong wind is pushing the boat, giving it an additional velocity of 2.0 m/s \(\hat{i}\) + 1.0 m/s \(\hat{j}\). (a) What is the velocity of the boat at t = 10 s? (b) What is the position of the boat at t = 10s? Draw a sketch of the boat’s trajectory and position at t = 10 s, showing the x- and y-axes.
    4. The position of a particle for t > 0 is given by \(\vec{r}\)(t) = (3.0t2 \(\hat{i}\) − 7.0t3 \(\hat{j}\) − 5.0t−2 \(\hat{k}\)) m. (a) What is the velocity as a function of time? (b) What is the acceleration as a function of time? (c) What is the particle’s velocity at t = 2.0 s? (d) What is its speed at t = 1.0 s and t = 3.0 s? (e) What is the average velocity between t = 1.0 s and t = 2.0 s?
    5. The acceleration of a particle is a constant. At t = 0 the velocity of the particle is (10 \(\hat{i}\) + 20 \(\hat{j}\))m/s. At t = 4 s the velocity is 10 \(\hat{j}\) m/s. (a) What is the particle’s acceleration? (b) How do the position and velocity vary with time? Assume the particle is initially at the origin.
    6. A particle has a position function \(\vec{r}\)(t) = cos(1.0t) \(\hat{i}\) + sin(1.0t) \(\hat{j}\) + t \(\hat{k}\), where the arguments of the cosine and sine functions are in radians. (a) What is the velocity vector? (b) What is the acceleration vector?
    7. A Lockheed Martin F-35 II Lighting jet takes off from an aircraft carrier with a runway length of 90 m and a takeoff speed 70 m/s at the end of the runway. Jets are catapulted into airspace from the deck of an aircraft carrier with two sources of propulsion: the jet propulsion and the catapult. At the point of leaving the deck of the aircraft carrier, the F-35’s acceleration decreases to a constant acceleration of 5.0 m/s2 at 30° with respect to the horizontal. (a) What is the initial acceleration of the F-35 on the deck of the aircraft carrier to make it airborne? (b) Write the position and velocity of the F-35 in unit vector notation from the point it leaves the deck of the aircraft carrier. (c) At what altitude is the fighter 5.0 s after it leaves the deck of the aircraft carrier? (d) What is its velocity and speed at this time? (e) How far has it traveled horizontally?

    Additional Problems

    1. Professional baseball player Nolan Ryan could pitch a baseball at approximately 160.0 km/h. At that average velocity, how long did it take a ball thrown by Ryan to reach home plate, which is 18.4 m from the pitcher’s mound? Compare this with the average reaction time of a human to a visual stimulus, which is 0.25 s.
    2. An airplane leaves Chicago and makes the 3000-km trip to Los Angeles in 5.0 h. A second plane leaves Chicago one-half hour later and arrives in Los Angeles at the same time. Compare the average velocities of the two planes. Ignore the curvature of Earth and the difference in altitude between the two cities.
    3. Unreasonable Results A cyclist rides 16.0 km east, then 8.0 km west, then 8.0 km east, then 32.0 km west, and finally 11.2 km east. If his average velocity is 24 km/ h, how long did it take him to complete the trip? Is this a reasonable time?
    4. An object has an acceleration of +1.2 cm/s2. At t = 4.0 s , its velocity is −3.4 cm/s. Determine the object’s velocities at t = 1.0 s and t = 6.0 s.
    5. A particle moves along the x-axis according to the equation x(t) = 2.0 − 4.0t2 m. What are the velocity and acceleration at t = 2.0 s and t = 5.0 s?
    6. A particle moving at constant acceleration has velocities of 2.0 m/s at t = 2.0 s and −7.6 m/s at t = 5.2 s. What is the acceleration of the particle?
    7. A train is moving up a steep grade at constant velocity (see following figure) when its caboose breaks loose and starts rolling freely along the track. After 5.0 s, the caboose is 30 m behind the train. What is the acceleration of the caboose?

    Figure shows a train moving up a hill.

    1. An electron is moving in a straight line with a velocity of 4.0 x 105 m/s. It enters a region 5.0 cm long where it undergoes an acceleration of 6.0 x 1012 m/s2 along the same straight line. (a) What is the electron’s velocity when it emerges from this region? b) How long does the electron take to cross the region?
    2. An ambulance driver is rushing a patient to the hospital. While traveling at 72 km/h, she notices the traffic light at the upcoming intersections has turned amber. To reach the intersection before the light turns red, she must travel 50 m in 2.0 s. (a) What minimum acceleration must the ambulance have to reach the intersection before the light turns red? (b) What is the speed of the ambulance when it reaches the intersection?
    3. A motorcycle that is slowing down uniformly covers 2.0 successive km in 80 s and 120 s, respectively. Calculate (a) the acceleration of the motorcycle and (b) its velocity at the beginning and end of the 2-km trip.
    4. A cyclist travels from point A to point B in 10 min. During the first 2.0 min of her trip, she maintains a uniform acceleration of 0.090 m/s2. She then travels at constant velocity for the next 5.0 min. Next, she decelerates at a constant rate so that she comes to a rest at point B 3.0 min later. (a) Sketch the velocity-versus-time graph for the trip. (b) What is the acceleration during the last 3 min? (c) How far does the cyclist travel?
    5. Two trains are moving at 30 m/s in opposite directions on the same track. The engineers see simultaneously that they are on a collision course and apply the brakes when they are 1000 m apart. Assuming both trains have the same acceleration, what must this acceleration be if the trains are to stop just short of colliding?
    6. A 10.0-m-long truck moving with a constant velocity of 97.0 km/h passes a 3.0-m-long car moving with a constant velocity of 80.0 km/h. How much time elapses between the moment the front of the truck is even with the back of the car and the moment the back of the truck is even with the front of the car?

    Top drawing shows passenger car with a speed of 80 kilometers per hour in front of the truck with the speed of 97 kilometers per hour. Middle drawing shows passenger car with a speed of 80 kilometers per hour parallel to the truck with the speed of 97 kilometers per hour. Bottom drawing shows passenger car with a speed of 80 kilometers per hour behind the truck with a speed of 97 kilometers per hour.

    1. A police car waits in hiding slightly off the highway. A speeding car is spotted by the police car doing 40 m/s. At the instant the speeding car passes the police car, the police car accelerates from rest at 4 m/s2 to catch the speeding car. How long does it take the police car to catch the speeding car?
    2. Pablo is running in a half marathon at a velocity of 3 m/s. Another runner, Jacob, is 50 meters behind Pablo with the same velocity. Jacob begins to accelerate at 0.05 m/s2. (a) How long does it take Jacob to catch Pablo? (b) What is the distance covered by Jacob? (c) What is the final velocity of the Jacob?
    3. Unreasonable results A runner approaches the finish line and is 75 m away; her average speed at this position is 8 m/s. She decelerates at this point at 0.5 m/s2. How long does it take her to cross the finish line from 75 m away? Is this reasonable?
    4. An airplane accelerates at 5.0 m/s2 for 30.0 s. During this time, it covers a distance of 10.0 km. What are the initial and final velocities of the airplane?
    5. Compare the distance traveled of an object that undergoes a change in velocity that is twice its initial velocity with an object that changes its velocity by four times its initial velocity over the same time period. The accelerations of both objects are constant.
    6. An object is moving east with a constant velocity and is at position x0 at time t0 = 0. (a) With what acceleration must the object have for its total displacement to be zero at a later time t? (b) What is the physical interpretation of the solution in the case for t → \(\infty\)?
    7. A ball is thrown straight up. It passes a 2.00-m-high window 7.50 m off the ground on its path up and takes 1.30 s to go past the window. What was the ball’s initial velocity?
    8. A coin is dropped from a hot-air balloon that is 300 m above the ground and rising at 10.0 m/s upward. For the coin, find (a) the maximum height reached, (b) its position and velocity 4.00 s after being released, and (c) the time before it hits the ground.
    9. A soft tennis ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.10 m. (a) Calculate its velocity just before it strikes the floor. (b) Calculate its velocity just after it leaves the floor on its way back up. (c) Calculate its acceleration during contact with the floor if that contact lasts 3.50 ms (3.50 x 10−3 s) (d) How much did the ball compress during its collision with the floor, assuming the floor is absolutely rigid?
    10. Unreasonable results. A raindrop falls from a cloud 100 m above the ground. Neglect air resistance. What is the speed of the raindrop when it hits the ground? Is this a reasonable number?
    11. Compare the time in the air of a basketball player who jumps 1.0 m vertically off the floor with that of a player who jumps 0.3 m vertically.
    12. Suppose that a person takes 0.5 s to react and move his hand to catch an object he has dropped. (a) How far does the object fall on Earth, where g = 9.8 m/s2? (b) How far does the object fall on the Moon, where the acceleration due to gravity is 1/6 of that on Earth?
    13. A hot-air balloon rises from ground level at a constant velocity of 3.0 m/s. One minute after liftoff, a sandbag is dropped accidentally from the balloon. Calculate (a) the time it takes for the sandbag to reach the ground and (b) the velocity of the sandbag when it hits the ground.
    14. (a) A world record was set for the men’s 100-m dash in the 2008 Olympic Games in Beijing by Usain Bolt of Jamaica. Bolt “coasted” across the finish line with a time of 9.69 s. If we assume that Bolt accelerated for 3.00 s to reach his maximum speed, and maintained that speed for the rest of the race, calculate his maximum speed and his acceleration. (b) During the same Olympics, Bolt also set the world record in the 200-m dash with a time of 19.30 s. Using the same assumptions as for the 100-m dash, what was his maximum speed for this race?
    15. An object is dropped from a height of 75.0 m above ground level. (a) Determine the distance traveled during the first second. (b) Determine the final velocity at which the object hits the ground. (c) Determine the distance traveled during the last second of motion before hitting the ground.
    16. A steel ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.45 m. (a) Calculate its velocity just before it strikes the floor. (b) Calculate its velocity just after it leaves the floor on its way back up. (c) Calculate its acceleration during contact with the floor if that contact lasts 0.0800 ms (8.00 x 10−5 s) (d) How much did the ball compress during its collision with the floor, assuming the floor is absolutely rigid?
    17. An object is dropped from a roof of a building of height h. During the last second of its descent, it drops a distance h/3. Calculate the height of the building.
    18. A Formula One race car is traveling at 89.0 m/s along a straight track enters a turn on the race track with radius of curvature of 200.0 m. What centripetal acceleration must the car have to stay on the track?
    19. A particle travels in a circular orbit of radius 10 m. Its speed is changing at a rate of 15.0 m/s2 at an instant when its speed is 40.0 m/s. What is the magnitude of the acceleration of the particle?
    20. The driver of a car moving at 90.0 km/h presses down on the brake as the car enters a circular curve of radius 150.0 m. If the speed of the car is decreasing at a rate of 9.0 km/h each second, what is the magnitude of the acceleration of the car at the instant its speed is 60.0 km/h?
    21. A race car entering the curved part of the track at the Daytona 500 drops its speed from 85.0 m/s to 80.0 m/s in 2.0 s. If the radius of the curved part of the track is 316.0 m, calculate the total acceleration of the race car at the beginning and ending of reduction of speed.
    22. An elephant is located on Earth’s surface at a latitude \(\lambda\). Calculate the centripetal acceleration of the elephant resulting from the rotation of Earth around its polar axis. Express your answer in terms of \(\lambda\), the radius RE of Earth, and time T for one rotation of Earth. Compare your answer with g for \(\lambda\) = 40°.

    The earth is illustrated rotating about the vertical north south axis. The equator is shown as a horizontal circle at the earth’s surface, centered on the earth’s center. A second circle at the earth’s surface, parallel to the equator but north of it, is shown. This circle is at latitude lambda, meaning that the angle between the radius to this circle and to the equator is lambda.

    1. A proton in a synchrotron is moving in a circle of radius 1 km and increasing its speed by v(t) = c1 + c2t2, where c1 = 2.0 x 105 m/s, c2 = 105 m/s3. (a) What is the proton’s total acceleration at t = 5.0 s? (b) At what time does the expression for the velocity become unphysical?
    2. A propeller blade at rest starts to rotate from t = 0 s to t = 5.0 s with a tangential acceleration of the tip of the blade at 3.00 m/s2. The tip of the blade is 1.5 m from the axis of rotation. At t = 5.0 s, what is the total acceleration of the tip of the blade?
    3. A particle is executing circular motion with a constant angular frequency of \(\omega\) = 4.00 rad/s. If time t = 0 corresponds to the position of the particle being located at y = 0 m and x = 5 m, (a) what is the position of the particle at t = 10 s? (b) What is its velocity at this time? (c) What is its acceleration?
    4. A particle’s centripetal acceleration is aC = 4.0 m/s2 at t = 0 s where it is on the x-axis and moving counterclockwise in the xy plane. It is executing uniform circular motion about an axis at a distance of 5.0 m. What is its velocity at t = 10 s?
    5. A rod 3.0 m in length is rotating at 2.0 rev/s about an axis at one end. Compare the centripetal accelerations at radii of (a) 1.0 m, (b) 2.0 m, and (c) 3.0 m.
    6. A particle located initially at (1.5 \(\hat{j}\) + 4.0 \(\hat{k}\))m undergoes a displacement of (2.5 \(\hat{i}\) + 3.2 \(\hat{j}\) − 1.2 \(\hat{k}\)) m. What is the final position of the particle?
    7. The position of a particle is given by \(\vec{r}\)(t) = (50 m/s)t \(\hat{i}\) − (4.9 m/s2)t2 \(\hat{j}\). (a) What are the particle’s velocity and acceleration as functions of time? (b) What are the initial conditions to produce the motion?
    8. A spaceship is traveling at a constant velocity of \(\vec{v}\)(t) = 250.0 \(\hat{i}\) m/s when its rockets fire, giving it an acceleration of \(\vec{a}\)(t) = (3.0 \(\vec{i}\) + 4.0 \(\hat{k}\))m/s2. What is its velocity 5 s after the rockets fire?
    9. A crossbow is aimed horizontally at a target 40 m away. The arrow hits 30 cm below the spot at which it was aimed. What is the initial velocity of the arrow?
    10. A long jumper can jump a distance of 8.0 m when he takes off at an angle of 45° with respect to the horizontal. Assuming he can jump with the same initial speed at all angles, how much distance does he lose by taking off at 30°?
    11. On planet Arcon, the maximum horizontal range of a projectile launched at 10 m/s is 20 m. What is the acceleration of gravity on this planet?
    12. A mountain biker encounters a jump on a race course that sends him into the air at 60° to the horizontal. If he lands at a horizontal distance of 45.0 m and 20 m below his launch point, what is his initial speed?
    13. Which has the greater centripetal acceleration, a car with a speed of 15.0 m/s along a circular track of radius 100.0 m or a car with a speed of 12.0 m/s along a circular track of radius 75.0 m?
    14. A geosynchronous satellite orbits Earth at a distance of 42,250.0 km and has a period of 1 day. What is the centripetal acceleration of the satellite?
    15. Two speedboats are traveling at the same speed relative to the water in opposite directions in a moving river. An observer on the riverbank sees the boats moving at 4.0 m/s and 5.0 m/s. (a) What is the speed of the boats relative to the river? (b) How fast is the river moving relative to the shore?

    Challenge Problems

    1. In a 100-m race, the winner is timed at 11.2 s. The second-place finisher’s time is 11.6 s. How far is the second-place finisher behind the winner when she crosses the finish line? Assume the velocity of each runner is constant throughout the race.
    2. The position of a particle moving along the x-axis varies with time according to x(t) = 5.0t2 − 4.0t3 m. Find (a) the velocity and acceleration of the particle as functions of time, (b) the velocity and acceleration at t = 2.0 s, (c) the time at which the position is a maximum, (d) the time at which the velocity is zero, and (e) the maximum position.
    3. A cyclist sprints at the end of a race to clinch a victory. She has an initial velocity of 11.5 m/s and accelerates at a rate of 0.500 m/s2 for 7.00 s. (a) What is her final velocity? (b) The cyclist continues at this velocity to the finish line. If she is 300 m from the finish line when she starts to accelerate, how much time did she save? (c) The second-place winner was 5.00 m ahead when the winner started to accelerate, but he was unable to accelerate, and traveled at 11.8 m/s until the finish line. What was the difference in finish time in seconds between the winner and runner-up? How far back was the runner-up when the winner crossed the finish line?
    4. In 1967, New Zealander Burt Munro set the world record for an Indian motorcycle, on the Bonneville Salt Flats in Utah, of 295.38 km/h. The one-way course was 8.00 km long. Acceleration rates are often described by the time it takes to reach 96.0 km/h from rest. If this time was 4.00 s and Burt accelerated at this rate until he reached his maximum speed, how long did it take Burt to complete the course?
    5. World’s Longest Par 3. The tee of the world’s longest par 3 sits atop South Africa’s Hanglip Mountain at 400.0 m above the green and can only be reached by helicopter. The horizontal distance to the green is 359.0 m. Neglect air resistance and answer the following questions. (a) If a golfer launches a shot that is 40° with respect to the horizontal, what initial velocity must she give the ball? (b) What is the time to reach the green?
    6. When a field goal kicker kicks a football as hard as he can at 45° to the horizontal, the ball just clears the 3-m-high crossbar of the goalposts 45.7 m away. (a) What is the maximum speed the kicker can impart to the football? (b) In addition to clearing the crossbar, the football must be high enough in the air early during its flight to clear the reach of the onrushing defensive lineman. If the lineman is 4.6 m away and has a vertical reach of 2.5 m, can he block the 45.7-m field goal attempt? (c) What if the lineman is 1.0 m away?

    The parabolic trajectory of a football is shown. A player kicks it up and to the right at an angle of theta to the horizontal. Another player to his right is jumping up but not quite reaching the trajectory. The trajectory passes through the goalposts to the right of both players.

    1. A truck is traveling east at 80 km/h. At an intersection 32 km ahead, a car is traveling north at 50 km/h. (a) How long after this moment will the vehicles be closest to each other? (b) How far apart will they be at that point?

    Contributors and Attributions

    Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).

    This page titled 3.E: Practice is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.