11.1.3.2: Explorations

Exploration 1: Addition of Displacement Vectors

Suppose that you use a radar system to track an airplane (the red circle) and the airplane travels according to the animation shown. Restart.

1. Draw a vector for the displacement of the airplane from \(t = 0\text{ s}\) to \(t = 8\text{ s}\). To do this, click the "Draw Vector" button. When a vector appears, drag it to the position of the airplane at \(t = 0\text{ s}\). Then play the animation, stop it at \(t = 8\text{ s}\), and adjust the tip of the vector until it is at this position.
2. Now draw a displacement vector for the airplane from \(t = 8\text{ s}\) to \(t = 16\text{ s}\). Use the same procedure as before. Be sure to click the "Draw Vector" button so that you can have a new vector to work with. You should see both the first displacement vector and the second displacement vector.
3. Now draw a displacement vector for the airplane from \(t = 0\text{ s}\) to \(t = 16\text{ s}\). Use the same procedure as before. What do you notice? To add vectors like this, you can connect the vectors from tail to head. The result, called the resultant vector, is the vector drawn from the tail of the first vector to the head of the last vector.
4. Click here to view the correct answer. How does your result compare to the correct answer?

Exploration authored by Aaron Titus with support by the National Science Foundation under Grant No. DUE-9952323 and placed in the public domain.

Exploration 2: Run the Gauntlet, Controlling \(x\), \(v\), and \(a\)

Drag the tip of the arrow to control the position, velocity, or acceleration of the object depending on which animation you choose.

Use the animation to answer the following questions (position is given in meters and time is given in seconds). Restart.

1. Can you navigate to the goal on the right? We call this challenge Running the Gauntlet.
2. Which controller (the position, velocity, or acceleration) is harder to use? Why?

Exploration 3: Acceleration of a Golf Ball That Rims the Hole

A putted golf ball "rims" the hole as shown in the animation (position is given in centimeters and time is given in seconds). Velocity vectors for the ball at the instant just before it hits the hole and the instant just after it hits the hole are shown. Restart. Note that the ball's speed does not change upon hitting the edge of the hole; this would not occur for an actual golf ball that rims the hole.

Suppose we want to find the average acceleration of the golf ball at some instant when it is in contact with the hole.

1. Draw the change-in-velocity vector using the velocity vectors shown. Click "Draw Vector" to add a vector to the animation and click "Clear Screen" to erase all drawn vectors.
2. What is the magnitude and direction of the change-in-velocity during this interval?
3. What is the average acceleration during this interval?
4. For the animation shown, at what instant do you think the instantaneous acceleration will equal the average acceleration of the golf ball during the time interval from \(0.9\text{ s}\) to \(1.2\text{ s}\)?
5. Click here to view the acceleration vector. If your change-in-velocity vector is still drawn on the screen, then you can stop the animation at the point where the acceleration vector and change-in-velocity vector match up. Did this occur at the instant you predicted?

Exploration authored by Aaron Titus with support by the National Science Foundation under Grant No. DUE-9952323 and placed in the public domain.

Exploration 4: Space Probe with Constant Acceleration

When you studied projectile motion, you learned that for projectile motion the \(x\) acceleration is zero and constant (which results in a constant \(x\) velocity) and the \(y\) acceleration is constant and downward toward Earth with a magnitude of \(9.8\text{ m/s}^{2}\). What mathematical curve describes the shape of the path of the projectile? Its shape is a parabola. It turns out that the shape of the path of any object that has constant acceleration and an initial velocity that is in a different direction than the acceleration is a parabola.

In the animation shown (position is given in meters and time is given in seconds), a space probe has engines that can fire on all four sides. Two of the engines engage at \(t = 2\text{ s}\). Restart. The acceleration is constant and zero before the engines engage, and it is constant (but not equal to zero) after the engines engage.

1. What is the direction of the \(x\) component of the acceleration after the engines engage?
2. What is the \(y\) velocity before the engines engage?
3. After the engines engage, how is the \(y\) velocity different?
4. Now click here to view the velocity and acceleration vectors. Do they match what you predicted?

Exploration authored by Aaron Titus with support by the National Science Foundation under Grant No. DUE-9952323 and placed in the public domain.

Exploration 5: Uphill and Downhill Projectile Motion

A projectile is launched at \(t = 0\text{ s}\) (position is given in meters and time is given in seconds). You may change the projectile's launch angle and initial speed and the height of the hill by using the text boxes and clicking the "set values and play" button. Restart.

For \(h = 0\text{ m}\), vary the projectile's launch angle and initial speed and consider the following questions.

1. For a given initial speed, what launch angle will provide the maximum range of the projectile?
2. For the value of launch angle in (a), what is the value of the initial speed that will hit the target?
3. What other value(s) of the projectile's launch angle and initial speed will enable the projectile to hit the target?
4. Are these values unique?
5. What is the general relationship between launch angle and initial speed?

For \(h = 10\text{ m}\), vary the projectile's launch angle and initial speed and consider the following questions.

6. For a given initial speed, what launch angle will provide the maximum horizontal displacement of the projectile?
7. What value(s) of the projectile's launch angle and initial speed will enable the projectile to hit the target?
8. Are these values unique?
9. Are these values the same as in (c)?

For \(h = -10\text{ m}\), vary the projectile's launch angle and initial speed and consider the following questions.

10. For a given initial speed, what launch angle will provide the maximum horizontal displacement of the projectile?
11. What value(s) of the projectile's launch angle and initial speed will enable the projectile to hit the target?
12. Are these values unique?
13. Are these values the same as in (c) and (g)?

Exploration 6: Uniform Circular Motion

A point (red) on a rotating wheel is shown in the animation (position is given in meters and time is given in seconds). Restart.

1. Note that the speed of the red point is constant. Is its velocity constant?
2. Click here to view the velocity vector. After viewing the vector rethink your answer: is the velocity of the red point constant?
3. What is the direction of the red point's acceleration vector? Click here to view the acceleration and velocity vectors.
4. How does the speed of the red point compare to the speed of another point, say a green one, which is at only half the radius of the red point? Click here to view both points. For clarity the green point is shown on the opposite side from the red one.
5. Why is the speed of the green point less than the speed of the red point?
6. How does the magnitude of the acceleration of the red point compare to the magnitude of the acceleration of the green point? Click here to view both points and their velocity and acceleration vectors.

Exploration authored by Aaron Titus with support by the National Science Foundation under Grant No. DUE-9952323 and placed in the public domain.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.