The moving golf ball shown in the diagram hits the golf club and comes to rest. What is the change in the ball’s momentum? Take the direction that the ball moves in as the positive direction.
This question is asking us about the change in the momentum of a golf ball. And if we look at the diagram, then we can see that we’ve got this golf ball moving towards a club. It’s labeled with a mass of 45 grams and a velocity of 20 centimeters per second to the left. Let’s call the mass of the golf ball 𝑚 and its velocity 𝑣 subscript i. The i here is used to indicate that this value of 20 centimeters per second is the initial velocity of the ball. We are told that this moving golf ball hits the golf club and comes to rest. This means that the golf ball’s velocity changes from this initial value of 20 centimeters per second to a final value of zero centimeters per second. We have labeled this final velocity as 𝑣 subscript f.
Another thing that’s important to notice here is that we’re told to take the positive direction as the direction that the ball is moving in. In the diagram, we can see that the golf ball is moving to the left, and so left must be our positive direction. Since left is our positive direction, then velocities to the left are positive and velocities to the right are negative. So, that’s why we’ve said that the ball’s initial velocity, which is directed to the left, has a value of positive 20 centimeters per second.
We are being asked to work out the change in the ball’s momentum. So, let’s recall that the momentum 𝑝 of an object is equal to the object’s mass 𝑚 multiplied by its velocity 𝑣. Now, just like velocity, momentum is a vector quantity, which means that as well as a magnitude, it also has a direction. The direction of an object’s momentum will be in the direction of its velocity. This means that a momentum in the leftward direction that the ball is initially moving will be positive, and a momentum in the opposite direction to the right will be negative.
As far as units are concerned, we typically want a mass measured in kilograms and a velocity in meters per second. This will give a momentum in units of kilogram-meters per second. Our value for the mass 𝑚 of the golf ball is in units of grams. To convert this to units of kilograms, we can recall that one kilogram is equal to 1000 grams, or equivalently, one gram is one thousandth of a kilogram. So, in kilograms, the mass 𝑚 is equal to 45 grams multiplied by one over 1000 kilograms per gram. Looking at the units, we can see that the grams cancel out and we’re left with units of kilograms. We then calculate a value of 0.045 kilograms.
We’re also going to want to convert our velocities from units of centimeters per second into units of meters per second. We know that one meter is equal to 100 centimeters. And so, one centimeter is one hundredth of a meter. So, we take our velocity in centimeters per second and multiply by one over 100 meters per centimeter. The centimeters and per centimeter cancel each other out, leaving us with units of meters per second. In these units, the initial velocity of the ball works out as 0.2 meters per second.
Now, we could do the same thing for the ball’s final velocity 𝑣 subscript f. But since this final velocity is when the ball is at rest, then we know that its value will be zero no matter what units we measure it in. So then, in units of meters per second, 𝑣 subscript f is equal to zero meters per second. Since we know the mass of the golf ball and we know its initial and final velocities, then we can substitute those values into this equation to calculate the ball’s initial and final momentum.
The initial momentum, which we’ve labeled as 𝑝 subscript i, is equal to the mass of 0.045 kilograms multiplied by 0.2 meters per second, which is the ball’s initial velocity. Evaluating the expression gives a result of 0.009 kilogram-meters per second. The ball’s final momentum 𝑝 subscript f is equal to 0.045 kilograms, that’s the ball’s mass, multiplied by its final velocity of zero meters per second. This gives it a final momentum of zero kilogram-meters per second.
We’re being asked to find the change in the momentum of the ball. And this change in momentum, which we’ve called Δ𝑝, must be equal to its final momentum 𝑝 subscript f minus its initial momentum 𝑝 subscript i. Let’s clear ourselves some space so that we can substitute our values for the initial and final momentum into this equation. When we substitute in our values, we get that Δ𝑝 is equal to the final momentum of zero kilogram-meters per second minus the initial momentum of 0.009 kilogram-meters per second. This works out as negative 0.009 kilogram-meters per second.
A negative change in momentum means that the amount of momentum in the positive direction that the golf ball has has decreased. And this makes sense since the ball started out moving in the positive leftward direction, which gave it a positive amount of initial momentum. The ball then ended up not moving at all, giving it a final momentum of zero. So, we can see that the golf ball has gone from having a momentum in the positive direction to having no momentum at all, which means that this negative change in momentum does make sense.
Our final answer, then, is that the change in the ball’s momentum is equal to negative 0.009 kilogram-meters per second.