Video Transcript
During a house removal, a wheelchair with a mass of 43.2 kilograms and a wheeled desk with a mass of 108 kilograms collide with each other. Each object applies to the other a force of 180 newtons and the collision lasts for 0.15 seconds. What is the magnitude of the momentum change of the chair? What is the magnitude of the momentum change of the desk? What is the magnitude of the velocity change of the chair? What is the magnitude of the velocity change of the desk?
Okay, so in this question we’re looking at the collision between a chair with wheels and a desk with wheels. Now we know that the chair has a mass 43.2 kilograms, and the desk has a mass of 108 kilograms. As well as this, we know that they exert on each other a force of 180 newtons. And finally, we also know that this collision lasts for 0.15 seconds.
Now the first thing that we need to do is to work out the magnitude or size of the momentum change of the chair. To do this, we need to recall how to calculate the impulse exerted on the chair. The impulse on an object is defined as the force exerted on an object 𝐹 multiplied by the short period of time for which that force acts on the object, which is called Δ𝑡. And we can also recall that the impulse on an object is equal to the change in momentum of the object, Δ𝑝.
So since we want to work out the momentum change of the chair in the first part of the question, we can simply work out the impulse on the chair. Now we can already work this out because we know the force that’s acting on the chair and the amount of time for which that force acts, because that force acts during the collision and the collision last 0.15 seconds.
So we can say that Δ𝑝 sub chair, or in other words the momentum change of the chair, is equal to 𝐹 sub chair, which is the force acting on the chair during the collision, multiplied by Δ𝑡, which is the time for which the collision occurs. At this point, we can substitute in the values. The force is 180 newtons, and the Δ𝑡, the time for which the force acts, is 0.15 seconds.
So we find that the change in momentum of the chair Δ𝑝 sub chair is equal to 27 newton seconds, because we had the force in newtons and Δ𝑡 was in seconds. So the unit that we’ll find is going to be newton seconds. However, we can recall that one newton is equal to one kilogram meter per second squared. So we replace the Newton with kilogram meter per second squared to give us kilogram meters per second squared times second.
Now this is the same as newton seconds, and we can see here that one of these powers of seconds cancels with one of these powers. And so the final unit that we have left is kilogram meters per second. And this unit makes sense because, remember, a momentum is defined as momentum is equal to mass multiplied by velocity. So mass has units of kilograms and velocity has units of meters per second. Hence, the units that we found here, kilogram meters per second, is the correct unit for the change in momentum of the chair.
And so we have our answer to the first part of the question. The magnitude of the momentum change of the chair is 27 kilograms meters per second. This means that we can move on now to working out the magnitude of the momentum change of the desk this time. We can do this using exactly the same method as earlier, using impulses. And this time we say that that Δ𝑝 sub desk, the change of momentum of the desk, is equal to the force on the desk, which is once again 180 newtons, multiplied by the time of the collision, which is 0.15 seconds still.
And when we evaluate the right-hand side of the equation, we’re once again going to get 27 kilograms meters per second. Now this makes physical sense. What we calculated in the first part of the question was the momentum change of the chair, but specifically the magnitude of the momentum change of the chair. And in this part of the question, we’ve calculated the magnitude of the momentum change of the desk.
Now we haven’t taken into account the directions in which these momentum changes occur. However, what’s happening in this collision is that due to conservation of momentum, any momentum lost by the chair is gained by the desk and vice versa. Therefore, if we just consider the magnitudes of the momentum changes of the desk and the chair, any momentum change in the chair is going to be the same magnitude as the momentum change in the desk, once again due to conservation of momentum.
And so the answer to the second part of the question is also 27 kilograms meters per second. Moving on then, we need to now find the magnitude of the velocity change of the chair. So we need to work out how much the velocity of the chair changes by, or specifically the size of the velocity change of the chair. To do this, we can recall that momentum is defined as mass multiplied by velocity. So a change in momentum is going to be defined as a change in mass multiplied by velocity.
However, in this collision, neither the chair nor the desk lose or gain any mass. In other words, their mass stays constant throughout the collision. Therefore, the only thing that can change in order to result in a change in momentum of the desk or the chair is the velocity of the desk or the chair. And therefore, we can say that the change in momentum of the desk or the chair is equal to the mass of the desk or the chair multiplied by the change in velocity of the desk or the chair.
We can then rearrange this equation by dividing both sides by the mass so that on the left-hand side we have the change in momentum divided by the mass and on the right we have the change in velocity. For the chair then, we can say that the change in velocity of the chair is equal to the change in momentum of the chair, which we calculated earlier was 27 kilograms meters per second as we’ve calculated over here, and we divide this by the mass of the chair which is 43.2 kilograms.
Now if we look closely at the units, in the numerator, we’ve got the units of momentum, kilograms meters per second; and in the denominator, we’ve got kilograms. So we’ve got kilograms meters per second divided by kilograms, and the kilograms cancel on the top and the bottom. This just leaves us with a unit of meters per second, which is a unit of velocity. And we’re calculating the change in velocity of the chair. So this makes sense.
When we evaluate the right-hand side of the equation then, we find that the change in velocity of the chair is 0.625 meters per second. And this is the final answer to the third part of the question, at which point we can move on to finding out the magnitude of the velocity change of the desk this time.
So we say that the change in velocity of the desk is the change in momentum of the desk, 27 kilograms meters per second as we found here, divided by the mass of desk, which is 108 kilograms. Evaluating this, we find that the change in velocity of the desk is 0.25 meters per second. So we have the final answer to the last part of the question. And at this point, we know that the magnitude of the momentum change of the chair is 27 kilograms meters per second. The magnitude of the momentum change of the desk is also 27 kilograms meters per second. The magnitude of the velocity change of the chair is 0.625 meters per second. And the magnitude of the velocity change of the desk is 0.25 meters per second.
