Exercise 1

The simulation below shows a ball rolling up a ramp. At regular intervals in time, it leaves a dot indicating its position at that time. This situation will form the basis for the next exercise.

The diagram below represents a strobe diagram of a ball as it rolls up a track. (In a strobe diagram, the position of an object is shown at instants separated by equal time intervals.)

On a large sheet of paper, draw vectors to represent the instantaneous velocity of the ball at each of the labeled locations. If the velocity is zero at any point, indicate that explicitly. Explain why you drew the vectors as you did.

We will call diagrams like this velocity diagrams. Unless otherwise specified, a velocity diagram shows the location of the velocity of an object at instants in time that are separated by equal time intervals.

On a sheet of paper, re-draw $\mathbf{v}_1$ and $\mathbf{v}_2$ with their tails together and use this to draw the vector that must be added to $\mathbf{v}_1$ to get $\mathbf{v}_2$. Label this vector $\Delta \mathbf{v}$ for change in velocity.

How does the direction of the change in velocity vector compare to the direction of the velocity vectors?

Would your answer change if you were to select two different consecutive points (e.g., points 3 and 4) while the ball was slowing down? Explain.

The graph below represents the velocity of the ball moving up the ramp.

How does the magnitude of the change in the velocity vector between points 1 and 2 compare to the magnitude of the change in the velocity vector between two different consecutive points (e.g., points 3 and 4)? Explain.

Consider the change in the velocity vector between two points that are not consecutive, e.g., points 1 and 4. Is the direction of the change in the velocity vector different than it was for consecutive points? Explain.

Is the magnitude of the change in velocity vector different than it was for consecutive points? Explain.

Use the definition of acceleration to draw and label a vector on your sheet of paper that represents the acceleration of the ball between points 1 and 2.

How is the direction of the acceleration vector related to the direction of the change in velocity vector? Explain.

Does the acceleration change as the ball rolls up the track? Would the acceleration vector you obtain differ if you were to choose (1) two different successive points on your diagram or (2) two points that are not consecutive? Explain.

Compare the direction of the acceleration and the direction of the velocity for an object that is moving in a straight line and slowing down. Explain.

Exercise 2

Now, we will let the ball roll down the ramp, starting from rest.

The diagram below represents a strobe diagram of a ball as it rolls down the same track.

Choose two consecutive points. On a sheet of paper, draw $\mathbf{v}_i$ and $\mathbf{v}_f$ with their tails together to represent the initial and final velocities of these two points. Use this to draw the vector that must be added to $\mathbf{v}_i$ to get $\mathbf{v}_f$. Label this vector $\Delta \mathbf{v}$ for change in velocity.

How does the direction of the change in velocity vector compare to the direction of the velocity vectors?

Would your answer change if you were to select two different consecutive points while the ball was speeding up? Explain.

Draw a vector on your sheet of paper that represents the acceleration of the ball between the two points chosen above.

How is the direction of the acceleration vector related to the direction of the change in velocity vector? Explain.

Compare the direction of the acceleration and the direction of the velocity for an object that is moving in a straight line and speeding up. Explain.

Exercise 3

Consider points 5 and 7 from above. On your sheet of paper, draw the velocity vectors and label them $\mathbf{v}_5$ and $\mathbf{v}_7$. Also draw and label the corresponding vector $\Delta \mathbf{v}_{57}$.

Now consider points 6 and 7. On your sheet of paper, draw the velocity vectors and label them $\mathbf{v}_6$ and $\mathbf{v}_7$. Also draw and label the corresponding vector $\Delta \mathbf{v}_{67}$.

How does the direction of $\Delta \mathbf{v}_{57}$ compare to the direction of $\Delta \mathbf{v}_{67}$? Explain.

Compare the direction of the acceleration of the ball at the turnaround point to that of the ball as it rolls: (1) up the track and (2) down the track?

Exercise 4

Below, the motion of the ball rolling up and down the incline from above is redrawn.

On a separate sheet of paper, sketch an acceleration diagram for the entire motion. (An acceleration diagram is similar to a velocity diagram; however the vectors on the acceleration diagram represent the acceleration rather than the velocity of the an object.)

Sketch x vs. t, v vs. t, and a vs. t graphs for the entire motion of a ball rolling up and then down the incline. Use a coordinate system in which the positive x-direction is down the track.

Sketch x vs. t, v vs. t, and a vs. t graphs for the entire motion of a ball rolling up and then down the incline. Use a coordinate system in which the positive x-direction is up the track.

Can an object have a negative acceleration and be speeding up? If so, describe a possible physical situation and a corresponding coordinate system. If not, explain why not.

Exercise 5

Carts A and B move along a horizontal track. The strobe diagram shows the locations of the carts at instants 1-5 separated by equal time intervals.

At instant 3 is cart A speeding up, slowing down, or moving with constant speed? Explain.

At instant 3 is cart B speeding up, slowing down, or moving with constant speed? Explain.

Is the speed of cart B greater than, less than, or equal to the speed of cart A at instant 2? Explain.

Is the speed of cart B greater than, less than, or equal to the speed of cart A at instant 3? Explain.

During a small time interval from just before instant 2 until just after instant 2, does the distance between cart A and cart B increase, decrease, or remain the same? Explain.

Is there any time interval during which cart A and cart B have the same average velocity? If so, identify the interval(s) and explain. If not, explain why not.

Is there any instant at which cart A and cart B have the same instantaneous velocity? If so, identify the instant(s) (e.g. “at instant 1,” or “at some instant between 2 and 3”) and explain. If not, explain why not.