# University of Wisconsin Green Bay

You drive a car in such a way that its motion is described by the velocity-time graph shown here. Draw the displacement-time and acceleration-time graphs that correspond to this motion, and describe in words how the car moves.

• In this problem, you are asked to describe the motion of the car. Whenever you are asked to describe the motion of an object without worrying about the cause of that motion, you have a kinematics problem. This problem is different from most kinematics problems, however, in that you are not asked for a numerical description but rather to use words and graphs to describe how the car moves.

• In this problem, the initial picture is provided for you. Continue to Select a Relation to draw the other two graphs.

• To go between a velocity-time graph and a displacement-time or acceleration-time graph, you need to understand how velocity, displacement and acceleration are related to each other. In other words, you need to use the definitions of velocity and acceleration:

v = Δx/Δt
In words, the value of velocity = the slope of the x-t graph.

a = Δv/Δt
In words, the value of acceleration = the slope of the v-t graph.

Hint: Graphing problems seem like they should be straightforward, and the equations that you need are only those given above. It is very, very common, however, to make mistakes on these problems because it feels like the graphs should be pictures of the motion and they are not. In order to avoid those mistakes, make a table based on the sentences above and then draw the graph from the table.

• Step 1:
Displacement-time graph

Once you understand the displacement-time graph, continue down to the acceleration-time graph.

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Step 2:
Acceleration-time graph

Once you understand the acceleration-time graph, proceed to the Understand page to describe the car’s motion in words.

• In this problem, you are given the velocity-time graph for the motion of a car. By relating the value of velocity to the slope of the x-t graph (this is just the definition of velocity) you are able to draw the x-t graph corresponding to this motion.

By relating the slope of the v-t graph to the value of acceleration (this is just the definition of acceleration) you are able to draw the a-t graph corresponding to this motion.

You can describe the motion looking at any of the three graphs.
From t = 0 to t1 : The car travels forward (+ direction) with a constant speed. There is no acceleration and the car moves away from its starting point at a constant rate.

From t1 to t2 : The car slows to a stop at a constant rate. It is still moving forward, but the amount of distance it covers in each second is decreasing. Acceleration acts against the motion of the car, or in the negative direction.

From t2 to t3 : The car reverses direction, moving faster and faster (at a constant acceleration) in the negative direction. Acceleration is acting with the motion of the car, so it is also in the negative direction. The amount of distance the car covers each second increases.

From t3 to t4 : The car continues to move in the negative direction but at a decreasing speed. The rate at which the speed decreases is getting greater—the driver is braking harder as the car stops—and so acceleration increases. Acceleration is acting against the motion of the car, or in the positive direction. The car covers less and less distance each second.