Question

A car traveling at 15 m/s passes a van that is starting from rest with an acceleration of 3 m/s². (a) How long will it take the van to catch up with the car? (b) How far has the car traveled when the van catches up with it?

Answer 1

Solution:

Coordinate of the car in the moment of time, when the van catches up with it:

$x_1=x_0+\upsilon_1\cdot{t}$

Coordinate of the van in the same moment of time:

$x_2=x_0+\frac{a_2\cdot{t^2}}{2}$

$x_1=x_2$ , so:

$x_0+\upsilon_1\cdot{t}=x_0+\frac{a_2\cdot{t^2}}{2}$

$\frac{a_2\cdot{t^2}}{2}-\upsilon_1\cdot{t}=0$

$t(\frac{a_2\cdot{t}}{2}-\upsilon_1)=0$

$t=0$ is the initial moment of time.

$\frac{a_2\cdot{t}}{2}=\upsilon_1$

So we can find the moment of time, when the van catches up with the car:

$t=\frac{2\cdot{\upsilon_1}}{a_2}=\frac{2\cdot{15}}{3}=10(s)$

And the coordinate of catching is:

$x=\upsilon_1\cdot{t}=15\cdot{10}=150(m)$

Answer: $t=10 s,\space{} x=150 m.$