Change in Momentum

Particles or objects can collide with other particles or objects, we know that this will often change their velocity (and maybe their mass) so their momentum is likely to change as well. We will deal with collisions in detail a little bit later but we are going to start by looking at the details of the change in momentum for a single particle or object.

Case 1: Object bouncing off a wall

Lets start with a simple picture, a ball of mass, \(m\), moving with initial velocity, \(\vec{v}_i\), to the right towards a wall. It will have momentum \(\vec{p}_i=m\vec{v}_i\) to the right as shown in this picture:

The ball bounces off the wall. It will now be moving to the left, with the same mass,but a different velocity, \(\vec{v}_f\) and therefore, a differentmomentum, \(\vec{p}_f=m\vec{v}_f\), as shown in this picture:

We know that the final momentum vector must be the sum of the initial momentumvector and the change in momentum vector, \(\Delta \vec{p}=m\Delta \vec{v}\).This means that, using tail-to-head vector addition, \(\Delta \vec{p}\), mustbe the vector that starts at the head of \(\vec{p}_i\) and ends on the head of\(\vec{p}_f\) as shown in this picture:

We also know from algebraic addition of vectors that:\begin{align*}\vec{p}_f &=\vec{p}_i + \Delta \vec{p} \\\vec{p}_f - \vec{p}_i &= \Delta \vec{p} \\\Delta \vec{p} &= \vec{p}_f - \vec{p}_i\end{align*}If we put this all together we can show the sequence and the change in momentumin one diagram:

We have just shown the case for a rebounding object. There are a few other cases wecan use to illustrate the basic features but they are all built up in the same way.

Case 2: Object stops

In some scenarios the object may come to a standstill (rest). An example of such a caseis a tennis ball hitting the net. The net stops the ball but doesn't cause it to bounce back.At the instant before it falls to the ground its velocity is zero. This scenario is describedin this image:

Case 3: Object continues more slowly

In this case, the object continue in the same direction but more slowly. To give thissome context, this could happen when a ball hits a glass window and goes throughit or an object sliding on a frictionless surface encounters a small rough patch before carryingon along the frictionless surface.

Important:

Note that even though the momentum remains in the same direction the change in momentum is in the opposite direction because the magnitude of the final momentum is less than the magnitude of the initial momentum.

Case 4: Object gets a boost

In this case the object interacts with something that increases the velocity it has without changing its direction. For example, in squash the ball can bounce off a back wall towards the front wall and a player can hit it with a racquet in the same direction, increasing its velocity.

If we analyse this scenario in the same way as the first 3 cases, it will look like this:

Case 5: Vertical bounce

Important:

For this explanation we are ignoring any effect of gravity. This isn't accurate butwe will learn more about the role of gravity in this scenarion in the next tutorial.

All of the examples that we've shown so far have been in the horizontal direction.That is just a coincidence, this approach applies for vertical or horizontal cases. In fact, it applies to any scenario where the initial and final vectors fall on the same line, any 1-dimensional (1D) problem. We will only deal with 1D scenarios in this tutorial. For example, a stationary basketball player bouncing a ball.

To illustrate the point, here is what the analysis would look like for a ballbouncing off the floor:

Change in Momentum