Problem 4.

A 2000 kg truck is traveling east through an intersection at 2 m/s when it is hit simultaneously from the side and the rear. One car is a 1000 kg compact traveling north at 5 m/s.  The other car is a 1500 kg midsize traveling east at 10 m/s.  The three vehicles become entangled and slide at one body.  What are their speeds and direction just after the collision?

Solution:

This is the example of perfectly inelastic collision: after the collision all parts of the system move as a whole. To find the final velocity we need to use the conservation of the net momentum.

The initial net momentum has three contributions:

- momentum of 2000 kg truck;

- momentum of 1000 kg compact;

- momentum of 1500 kg midsize.

We show these vehicles and the directions of the corresponding velocities (before the collision).

Then the (initial) net momentum is

In the final state all vehicles move with the same velocity. Then the final momentum is

Then momentum conservation takes the form

We need to rewrite this vector equation in terms of components. We introduce axis x - east direction, and axis y – north direction. Then the x- component of the above equation is

Then

The y- component of the above vector equation is

Then

Now we know x and y components of the final velocity. Then the magnitude of the final velocity is

The direction of the final velocity is characterized by the angle between the direction of velocity and axis x (east):