What Velocity vs. Time Graphs Can Tell You| University Physics

A common graphical representation of motion along a straight line is the v vs. t graph, that is, the graph of (instantaneous) velocity as a function of time. In this graph, time t is plotted on the horizontal axis and velocity v on the vertical axis. Note that by definition, velocity and acceleration are vector quantities. In a straight-line motion, however, these vectors have only a single nonzero component in the direction of motion. Thus, in this problem, we will call v the velocity and a the acceleration, even though they are really the components of the velocity and acceleration vectors in the direction of motion, respectively.

Here is a plot of velocity versus time for a particle that travels along a straight line with varying velocity. Refer to this plot to answer the following questions.

The plot shows the x component of the velocity as a function of time. Time is plotted from 0 to 50 seconds on the x-axis. The x component of the velocity is plotted from 0 to 2 meters per second on the y-axis. The velocity starts at 0 seconds and 0.5 meters per second, then increases linearly to 2.0 meters per second at 20 seconds and stays at this value until 40 seconds. Then it decreases linearly to 0 at 50 seconds.

PART A. What is the initial velocity of the particle, v_0?

ANSWER: v_0=0.5\:m/s

The initial velocity is the velocity at t=0s. Recall that in a graph of velocity versus time, time is plotted on the horizontal axis and velocity on the vertical axis.

PART B. What is the total distance Δx traveled by the particle?

ANSWER: \Delta x=75\:m

Recall that the area of the region that extends over a time interval Δt under the v vs. t curve is always equal to the distance traveled in Δt. Thus, to calculate the total distance, you need to find the area of the entire region under the v vs. t curve. In the case at hand, the entire region under the v vs. t curve is not an elementary geometrical figure, but rather a combination of triangles and rectangles.

PART C. What is the average acceleration a_{av} of the particle over the first 20.0 seconds?

ANSWER: a_{av}=0.075\:m/s^2

The average acceleration of a particle between two instants of time is the slope of the line connecting the two corresponding points in a v vs. t graph.

PART D. What is the instantaneous acceleration a of the particle at t=45.0\:s seconds?

ANSWER: a=-0.20\:m/s^2

The instantaneous acceleration of a particle at any point on a v vs. t graph is the slope of the line tangent to the curve at that point. Since in the last 10 seconds of motion, between t=40.0s and t=50.0s, the curve is a straight line, the tangent line is the curve itself. Physically, this means that the instantaneous acceleration of the particle is constant over that time interval. This is true for any motion where velocity increases linearly with time.

PART E. Which of the graphs shown below is the correct acceleration vs. time plot for the motion described in the previous parts?

There are four graphs showing the acceleration as a function of time. Time ranges from 0 to 50 seconds on the horizontal axis. The acceleration is from -0.20 to 0.2 meters per second squared on the vertical axis. On graph A, the acceleration equals 0.075 meters per second squared through the first 40 seconds, then equals -0.20 meters per second squared through the next 10 seconds. On graph B, the acceleration equals 0.075 meters per second squared through the first 20 seconds, then equals zero through the next 20 seconds and then equals -0.20 meters per second squared through the next 10 seconds. On graph C, the acceleration equals 0.2 meters per second squared through the first 20 seconds, then equals zero through the next 20 seconds. Then, it decreases linearly to -0.20 meters per second squared through the next 10 seconds. On graph D, the acceleration equals -0.075 meters per second squared through the first 20 seconds, then equals zero through the next 20 seconds and then equals to 0.075 meters per second squared through the next 10 seconds.

ANSWER: Graph B

Recall that whenever velocity increases linearly with time, acceleration is constant. In the example here, the particle’s velocity increases linearly with time in the first 20.0 s of motion. In the second 20.0 s , the particle’s velocity is constant, and then it decreases linearly with time in the last 10 s. This means that the particle’s acceleration is constant over each time interval, but its value is different in each interval.

In conclusion, graphs of velocity as a function of time are a useful representation of straight-line motion. If read correctly, they can provide you with all the information you need to study the motion.