A car accelerates from rest in a straight line with a constant acceleration. At the same time a cyclist passes the car in the same direction at constant speed v. What is the velocity of the car when it reaches the cyclist?

Image Credit:CC/Flickr/Richard Masoner/Cyclelicious

Solution.

The equation of the motion of the object

x_{f} = x_{i} + v_{i}∙t + a∙t^2/2

Let’s write the position of the car as a function of time

x_{car} = a∙t^2/2

Now we take in consideration the fact that the cyclist moves at constant speed v and we write the position of the cyclist as a function of time

x_{cyclist} = v∙t

If the car reaches the cyclist then

x_{car} = x_{cyclist}

Hence,

a∙t^2/2 = v∙t

and finally,

t = 2∙v/a (1)

The equation for the velocity as a function of time is given by the formula

v_{f} = v_{i} + a∙t

We remember that the car accelerates from rest, so the velocity of the car as a function of time is

v_{car} = a∙t (2)

Now we substitute the equation (1) into the equation (2) :

v_{car} = a∙2∙v/a

We simplify and we get

v_{car} = 2∙v