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16.4: Examples

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    • Christopher Duston, Merrimack College
    • OpenStax
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    Whiteboard Problem \(\PageIndex{1}\): The Range Formula

    The range of a projectile which starts and ends at the same altitude can be found with
    \[R=\frac{2v_0^2\sin\theta\cos\theta}{g},\]
    where \(v_0\) is the initial speed and \(\theta\) is the launch angle.

    1. Derive this formula from the kinematic equations for projectile motion.
    2. Find the angle at which this range will be maximized. Hint: Use the trig identify

    \[\sin 2\theta=2\sin\theta\cos\theta.\]

     

    Problem-Solving Strategy: Projectile Motion
    1. Resolve the motion into horizontal and vertical components along the x- and y-axes. The magnitudes of the components of displacement \(\vec{s}\) along these axes are x and y. The magnitudes of the components of velocity \(\vec{v}\) are vx = vcos\(\theta\) and vy = vsin\(\theta\), where v is the magnitude of the velocity and \(\theta\) is its direction relative to the horizontal, as shown in Figure \(\PageIndex{2}\).
    2. Treat the motion as two independent one-dimensional motions: one horizontal and the other vertical. Use the kinematic equations for horizontal and vertical motion presented earlier.
    3. Solve for the unknowns in the two separate motions: one horizontal and one vertical. Note that the only common variable between the motions is time t. The problem-solving procedures here are the same as those for one-dimensional kinematics and are illustrated in the following solved examples.
    4. Recombine quantities in the horizontal and vertical directions to find the total displacement \(\vec{s}\) and velocity \(\vec{v}\). Solve for the magnitude and direction of the displacement and velocity using \(s = \sqrt{x^{2} + y^{2}} \ldotp \quad \phi = \tan^{-1} \left(\dfrac{y}{x}\right), \quad v = \sqrt{v_{x}^{2} + v_{y}^{2}} \ldotp\)where \(\phi\) is the direction of the displacement \(\vec{s}\).
    Figure a shows the locations and velocities of a projectile on an x y coordinate system at 10 instants in time. When the projectile is at the origin, it has a velocity v sub 0 y which makes an angle theta sub 0 with the horizontal. The velocity is shown as a dark blue arrow, and its x and y components are shown as light blue arrow. The projectile’s position follows a downward-opening parabola, moving up to a maximum height, then back to y = 0, and continuing below the x axis The velocity, V, at each time makes an angle theta which changes in time, and has x component V sub x and y component v sub y. The x component of the velocity V sub x is the same at all times. The y component v sub y points up but gets smaller, until the projectile reaches the maximum height, where the velocity is horizontal and has no y component. After the maximum height, the velocity has a y component pointing down and growing larger. As the projectile reaches the same elevation on the way down as it had on the way up, its velocity is below the horizontal by the same angle theta as it was above the horizontal on the way up. In particular, when it comes back to y = 0 on the way down, the angle between the vector v and the horizontal is minus s=theta sub zero and the y component of the velocity is minus v sub 0 y. The last position shown is below the x axis, and the y component of the velocity is larger than it was initially. The graph clearly shows that the horizontal distances travelled in each of the time intervals are equal, while the vertical distances decrease on the way up and increase on the way down. Figure b shows the horizontal component, constant velocity. The horizontal positions and x components of the velocity of the projectile are shown along a horizontal line. The positions are evenly spaced, and the x components of the velocities are all the same, and point to the right. Figure c shows the vertical component, constant acceleration. The vertical positions and y components of the velocity of the projectile are shown along a vertical line. The positions are get closer together on the way up, then further apart on the way down. The y components of the velocities initially point up, decreasing in magnitude until there is no y component to the velocity at the maximum height. After the maximum height, the y components of the velocities point down and increase in magnitude. Figure d shows that putting the horizontal and vertical components of figures b and c together gives the total velocity at a point. The velocity V has an x component of V sub x, has y component of V sub y, and makes an angle of theta with the horizontal. In the example shown, the velocity has a downward y component.
    Figure \(\PageIndex{2}\): (a) We analyze two-dimensional projectile motion by breaking it into two independent one-dimensional motions along the vertical and horizontal axes. (b) The horizontal motion is simple, because ax = 0 and vx is a constant. (c) The velocity in the vertical direction begins to decrease as the object rises. At its highest point, the vertical velocity is zero. As the object falls toward Earth again, the vertical velocity increases again in magnitude but points in the opposite direction to the initial vertical velocity. (d) The x and y motions are recombined to give the total velocity at any given point on the trajectory.
    Example \(\PageIndex{2}\): A Fireworks Projectile Explodes high and away

    During a fireworks display, a shell is shot into the air with an initial speed of 70.0 m/s at an angle of 75.0° above the horizontal, as illustrated in Figure \(\PageIndex{3}\). The fuse is timed to ignite the shell just as it reaches its highest point above the ground. (a) Calculate the height at which the shell explodes. (b) How much time passes between the launch of the shell and the explosion? (c) What is the horizontal displacement of the shell when it explodes? (d) What is the total displacement from the point of launch to the highest point?

    The trajectory of a fireworks shell from its launch to its highest point is shown as the left half of a downward-opening parabola in a graph of y as a function of x. The maximum height is h = 233 meters and its x displacement at that time is x = 125 meters. The initial velocity vector v sub 0 is up and to the right, tangent to the trajectory curve, and makes an angle of theta sub 0 equal to 75 degrees.
    Figure \(\PageIndex{3}\): The trajectory of a fireworks shell. The fuse is set to explode the shell at the highest point in its trajectory, which is found to be at a height of 233 m and 125 m away horizontally.

    Strategy

    The motion can be broken into horizontal and vertical motions in which ax = 0 and ay = −g. We can then define x0 and y0 to be zero and solve for the desired quantities.

    Solution

    1. By “height” we mean the altitude or vertical position y above the starting point. The highest point in any trajectory, called the apex, is reached when vy = 0. Since we know the initial and final velocities, as well as the initial position, we use the following equation to find y: \(v_{y}^{2} = v_{0y}^{2} - 2g(y - y_{0}) \ldotp\)Because y0 and vy are both zero, the equation simplifies to \(0 = v_{0y}^{2} - 2gy \ldotp\)Solving for y gives \(y = \frac{v_{0y}^{2}}{2g} \ldotp\)Now we must find v0y, the component of the initial velocity in the y direction. It is given by v0y = v0 sin\(\theta_{0}\), where v0 is the initial velocity of 70.0 m/s and \(\theta_{0}\) = 75° is the initial angle. Thus \(v_{0y} = v_{0} \sin \theta = (70.0\; m/s) \sin 75^{o} = 67.6\; m/s\)and y is \(y = \frac{(67.6\; m/s)^{2}}{2(9.80\; m/s^{2})} \ldotp\)Thus, we have \(y = 233\; m \ldotp\)Note that because up is positive, the initial vertical velocity is positive, as is the maximum height, but the acceleration resulting from gravity is negative. Note also that the maximum height depends only on the vertical component of the initial velocity, so that any projectile with a 67.6-m/s initial vertical component of velocity reaches a maximum height of 233 m (neglecting air resistance). The numbers in this example are reasonable for large fireworks displays, the shells of which do reach such heights before exploding. In practice, air resistance is not completely negligible, so the initial velocity would have to be somewhat larger than that given to reach the same height.
    2. As in many physics problems, there is more than one way to solve for the time the projectile reaches its highest point. In this case, the easiest method is to use vy = v0y − gt. Because vy = 0 at the apex, this equation reduces \(0 = v_{0y} - gt\)or \(t = \frac{v_{0y}}{g} = \frac{67.6\; m/s}{9.80\; m/s^{2}} = 6.90\; s \ldotp\)This time is also reasonable for large fireworks. If you are able to see the launch of fireworks, notice that several seconds pass before the shell explodes. Another way of finding the time is by using y = y0 + \(\frac{1}{2}\)(v0y + vy)t. This is left for you as an exercise to complete.
    3. Because air resistance is negligible, ax = 0 and the horizontal velocity is constant, as discussed earlier. The horizontal displacement is the horizontal velocity multiplied by time as given by x = x0 + vxt, where x0 is equal to zero. Thus, \(x = v_{x} t,\)where vx is the x-component of the velocity, which is given by \(v_{x} = v_{0} \cos \theta = (70.0\; m/s) \cos 75^{o} = 18.1\; m/s \ldotp\)Time t for both motions is the same, so x is \(x = (18.1\; m/s)(6.90\; s) = 125\; m \ldotp\)Horizontal motion is a constant velocity in the absence of air resistance. The horizontal displacement found here could be useful in keeping the fireworks fragments from falling on spectators. When the shell explodes, air resistance has a major effect, and many fragments land directly below.
    4. The horizontal and vertical components of the displacement were just calculated, so all that is needed here is to find the magnitude and direction of the displacement at the highest point: \(\vec{s} = 125 \hat{i} + 233 \hat{j}\)\(|\vec{s}| = \sqrt{125^{2} + 233^{2}} = 264\; m\)\(\theta = \tan^{-1} \left(\dfrac{233}{125}\right) = 61.8^{o} \ldotp\)Note that the angle for the displacement vector is less than the initial angle of launch. To see why this is, review Figure \(\PageIndex{1}\), which shows the curvature of the trajectory toward the ground level. When solving Example 4.7(a), the expression we found for y is valid for any projectile motion when air resistance is negligible. Call the maximum height y = h. Then, \(h = \frac{v_{0y}^{2}}{2g} \ldotp\)This equation defines the maximum height of a projectile above its launch position and it depends only on the vertical component of the initial velocity.
    Exercise \(\PageIndex{3}\)

    A rock is thrown horizontally off a cliff 100.0 m high with a velocity of 15.0 m/s. (a) Define the origin of the coordinate system. (b) Which equation describes the horizontal motion? (c) Which equations describe the vertical motion? (d) What is the rock’s velocity at the point of impact?

    Example \(\PageIndex{4}\): Calculating projectile motion- Tennis Player

    A tennis player wins a match at Arthur Ashe stadium and hits a ball into the stands at 30 m/s and at an angle 45° above the horizontal (Figure \(\PageIndex{4}\)). On its way down, the ball is caught by a spectator 10 m above the point where the ball was hit. (a) Calculate the time it takes the tennis ball to reach the spectator. (b) What are the magnitude and direction of the ball’s velocity at impact?

    An illustration of a tennis ball launched into the stands. The player is to the left of the stands and hits the ball up and to the right at an angle of theta equal to 45 degrees and velocity of v sub 0 equal to 30 meters per second. The ball reaches a spectator who is seated 10 meters above the initial height of the ball.
    Figure \(\PageIndex{4}\): The trajectory of a tennis ball hit into the stands.

    Strategy

    Again, resolving this two-dimensional motion into two independent one-dimensional motions allows us to solve for the desired quantities. The time a projectile is in the air is governed by its vertical motion alone. Thus, we solve for t first. While the ball is rising and falling vertically, the horizontal motion continues at a constant velocity. This example asks for the final velocity. Thus, we recombine the vertical and horizontal results to obtain \(\vec{v}\) at final time t, determined in the first part of the example.

    Solution

    1. While the ball is in the air, it rises and then falls to a final position 10.0 m higher than its starting altitude. We can find the time for this by using Equation \ref{4.22}: \(y = y_{0} + v_{0y}t - \frac{1}{2} gt^{2} \ldotp\)If we take the initial position y0 to be zero, then the final position is y = 10 m. The initial vertical velocity is the vertical component of the initial velocity: \(v_{0y} = v_{0} \sin \theta_{0} = (30.0\; m/s) \sin 45^{o} = 21.2\; m/s \ldotp\)Substituting into Equation \ref{4.22} for y gives us \(10.0\; m = (21.2\; m/s)t − (4.90\; m/s^{2})t^{2} \ldotp\)Rearranging terms gives a quadratic equation in t: \((4.90\; m/s^{2})t^{2} − (21.2\; m/s)t + 10.0\; m = 0 \ldotp\)Use of the quadratic formula yields t = 3.79 s and t = 0.54 s. Since the ball is at a height of 10 m at two times during its trajectory—once on the way up and once on the way down—we take the longer solution for the time it takes the ball to reach the spectator: \(t = 3.79\; s \ldotp\)The time for projectile motion is determined completely by the vertical motion. Thus, any projectile that has an initial vertical velocity of 21.2 m/s and lands 10.0 m above its starting altitude spends 3.79 s in the air.
    2. We can find the final horizontal and vertical velocities vx and vy with the use of the result from (a). Then, we can combine them to find the magnitude of the total velocity vector \(\vec{v}\) and the angle \(\theta\) it makes with the horizontal. Since vx is constant, we can solve for it at any horizontal location. We choose the starting point because we know both the initial velocity and the initial angle. Therefore, \(v_{x} = v_{0} \cos \theta_{0} = (30\; m/s) \cos 45^{o} = 21.2\; m/s \ldotp\)The final vertical velocity is given by Equation \ref{4.21}: \(v_{y} = v_{0y} − gt \ldotp\)Since \(v_{0y}\) was found in part (a) to be 21.2 m/s, we have \(v_{y} = 21.2\; m/s − (9.8\; m/s^{2})(3.79 s) = −15.9\; m/s \ldotp\)The magnitude of the final velocity \(\vec{v}\) is \(v = \sqrt{v_{x}^{2} + v_{y}^{2}} = \sqrt{(21.2\; m/s)^{2} + (-15.9\; m/s)^{2}} = 26.5\; m/s \ldotp\)The direction \(\theta_{v}\) is found using the inverse tangent: \(\theta_{v} = \tan^{-1} \left(\dfrac{v_{y}}{v_{x}}\right) = \tan^{-1} \left(\dfrac{21.2}{-15.9}\right) = -53.1^{o} \ldotp\)

    Significance

    1. As mentioned earlier, the time for projectile motion is determined completely by the vertical motion. Thus, any projectile that has an initial vertical velocity of 21.2 m/s and lands 10.0 m above its starting altitude spends 3.79 s in the air.
    2. The negative angle means the velocity is 53.1° below the horizontal at the point of impact. This result is consistent with the fact that the ball is impacting at a point on the other side of the apex of the trajectory and therefore has a negative y component of the velocity. The magnitude of the velocity is less than the magnitude of the initial velocity we expect since it is impacting 10.0 m above the launch elevation.
    Example \(\PageIndex{5}\): Comparing golf shots

    A golfer finds himself in two different situations on different holes. On the second hole he is 120 m from the green and wants to hit the ball 90 m and let it run onto the green. He angles the shot low to the ground at 30° to the horizontal to let the ball roll after impact. On the fourth hole he is 90 m from the green and wants to let the ball drop with a minimum amount of rolling after impact. Here, he angles the shot at 70° to the horizontal to minimize rolling after impact. Both shots are hit and impacted on a level surface. (a) What is the initial speed of the ball at the second hole? (b) What is the initial speed of the ball at the fourth hole? (c) Write the trajectory equation for both cases. (d) Graph the trajectories.

    Strategy

    We see that the range equation has the initial speed and angle, so we can solve for the initial speed for both (a) and (b). When we have the initial speed, we can use this value to write the trajectory equation.

    Solution

    1. \(R = \frac{v_{0}^{2} \sin 2 \theta_{0}}{g} \Rightarrow v_{0} = \sqrt{\dfrac{Rg}{\sin 2\theta_{0}}} = \sqrt{\dfrac{(90.0\; m)(9.8\; m/s^{2})}{\sin (2(30^{o}))}} = 31.9\; m/s\)
    2. \(R = \frac{v_{0}^{2} \sin 2 \theta_{0}}{g} \Rightarrow v_{0} = \sqrt{\dfrac{Rg}{\sin 2\theta_{0}}} = \sqrt{\dfrac{(90.0\; m)(9.8\; m/s^{2})}{\sin (2(70^{o}))}} = 37.0\; m/s\)
    3. \(y = x \Big[ \tan \theta_{0} - \frac{g}{2(v_{0} \cos \theta_{0})^{2}} x \Big]\)Second hole: \(y = x \Big[ \tan 30^{o} - \frac{9.8\; m/s^{2}}{2[(31.9\; m/s)(\cos 30^{o})]^{2}} x \Big] = 0.58x - 0.0064x^{2}\)Fourth hole: \(y = x \Big[ \tan 70^{o} - \frac{9.8\; m/s^{2}}{2[(37.0\; m/s)(\cos 70^{o})]^{2}} x \Big] = 2.75x - 0.0306x^{2}\)
    4. Using a graphing utility, we can compare the two trajectories, which are shown in Figure \(\PageIndex{6}\).
    Two parabolic functions are shown. The range for both trajectories is 90 meters. One shot travels much higher than the other. The higher shot has an initial velocity of 37 meters per second and an angle of 70 degrees. The lower shot has an initial velocity of 31.9 meters per second and an angle of 30 degrees.
    Figure \(\PageIndex{6}\): Two trajectories of a golf ball with a range of 90 m. The impact points of both are at the same level as the launch point.

    Significance

    The initial speed for the shot at 70° is greater than the initial speed of the shot at 30°. Note from Figure \(\PageIndex{6}\) that two projectiles launched at the same speed but at different angles have the same range if the launch angles add to 90°. The launch angles in this example add to give a number greater than 90°. Thus, the shot at 70° has to have a greater launch speed to reach 90 m, otherwise it would land at a shorter distance.

    Exercise \(\PageIndex{6}\)

    If the two golf shots in Example 4.9 were launched at the same speed, which shot would have the greatest range?

    Simulation

    At PhET Explorations: Projectile Motion, learn about projectile motion in terms of the launch angle and initial velocity.

    This page titled 16.4: Examples is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Christopher Duston, Merrimack College (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.