Collision Problems
When one body hits another, they velocities change. In simple cases, we may be able to calculate the final velocities. This is possible if we know the masses and the initial velocities, and one or two other conditions.
The main equation that we need is the conservaton of momentum:
where
m1 = mass of body 1
u1 = initial velocity of body 1
v1 = final velocity of body 1
m2 = mass of body 2
u2 = initial velocity of body 2
v2 = final velocity of body 2
To see how this is used, lets look at some examples.
Exercise. A stone flying at 5 m/s hits a book resting on a smooth table, and bounces back at 1 m/s. The mass of the stone is 100 g, abd the book is 300 g. At what velocity does the book take off?
Answer. A direct substitution into the above equation gives
I have converted 300 g to 0.3 kg, and 100 g to 0.1 kg. The initial velocity of the book is 0 m/s, because it is initially at rest. I take the initial direction of the stone as the positive direction, so I need to add a minus sign to its final velocity since it bounces back. Solving gives v2 = 2 m/s as the final velocity of the book.
Note that we only consider motion along a straight line. For motion in any direction, the velocities in the above equation must be replaced by vectors. We shall not go into this.
Exercise. A boy on a boat jumps into the sea. The boy is 20 kg, and the mass of the boat is 10 kg. If the boy takes off at 2 m/s, find the velocity of the boat after the jump.
Answer. Applying the above equation gives
We take it that just before the jump, the boat with the boy is at rest, so the initial velocities are zeros. We take the direction of jump as the positive direction, so the final velocity of the boy is +2 m/s. Solving, we find that the final velocity of the boat is v2 = -4 m/s. The minus sign means that the boat moves in the opposite direction to the boy. This is expected, since the boy had to push back on the boat in order to jump.
Copyright 2011 by Kai Hock. All rights reserved.
Last updated: 10 March 2011