How to find velocity when given a linear graph and what t equals?

Question

A car travels along a straight road for 30 seconds starting at time t = 0. Its acceleration in ft/sec² is given by the linear graph below for the time interval [0, 30]. At t = 0, the velocity of the car is 0 and its position is 10.

What is the velocity of the car when t = 6? You must show your work and include units in your answer.

Answer 1

Hey Joshua!

This is a bit hard to answer without the graph but I'll give you some tips to approach the problem. The problem gives you a linear graph that represents the acceleration of the car. This means that the graph can be converted to a formula of the form:

a(t) = mt + b

You also have the conditions:

v(0) = 0

and

s(0) = 10

where v is the velocity in terms of time and s is the position in terms of time. The trick to these problems is that acceleration is the derivative of velocity and velocity is the derivative of position. So to solve for velocity you want to integrate your formula for acceleration which you get from your graph. This will give you something of the form: v(t) = (m/2)t² + bt + c. You can then use the fact that v(0) = 0 to solve for c. It should then be simple enough to plug in t=6 to find the velocity of the car. Hope that helps!