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

### 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.