A tennis racquet hits a tennis ball that has a mass of 60.5 grams and applies a constant 75-newton force to it. The tennis ball changes its velocity by 30 meters per second during the collision. For how many seconds does the collision last?
Okay, so to answer this question, let’s start out by drawing a diagram. So here’s a rather badly drawn side-on view of a tennis racquet. And here’s our tennis racquet hitting a tennis ball. Now let’s say that this tennis ball has a mass 𝑚.
We’ve been told in the question that 𝑚 is equal to 60.5 grams. As well as this, let’s say that the tennis racquet exerts a force 𝐹 on the ball. Once again, we’ve been told in the question that the value of 𝐹 is 75 newtons. Let us also say that the change in velocity of the ball, which is caused by the collision with the racquet, is Δ𝑣. And we know once again from the question that Δ𝑣 is 30 meters per second.
What we’ve been asked to do is to find out the amount of time or for how many seconds the collision between the tennis ball and the tennis racquet lasts. In other words, let’s say that Δ𝑡, the time interval for which the collision between the racquet and the ball lasts, is as of yet unknown to us.
Now we need to go about working out the value of Δ𝑡. To do this, we can recall that an impulse on an object is defined as the force applied on an object multiplied by the time interval, Δ𝑡, for which that force acts on the object. So in this case, our object is the tennis ball. And a tennis racquet is applying a force 𝐹 onto the tennis ball for a time interval Δ𝑡, which is the amount of time for which the collision between the ball and the racquet occurs.
Now the other thing that we can also recall is that the impulse is equal to the change in momentum of the object, Δ𝑝. At this point, however, we don’t know what Δ𝑝 is. But we can recall that the momentum of an object is given by the mass of the object multiplied by the velocity of the object. So we can recall that the change in momentum, Δ𝑝, of that object is given as the change in mass times velocity of the object.
However, in this case, our object is the tennis ball. And the mass of that tennis ball is not changing during the collision. So we can say that the change in momentum of the ball is equal to 𝑚 multiplied by Δ𝑣. This is because only the velocity changes at all. And remember, this expression is only true for when the mass of the object stays constant.
Now at this point, we can work out the value of Δ𝑝. Δ𝑝 is equal to the mass of the ball, which is 60.5 grams. But let’s not use grams. Let’s convert it to the standard unit of kilograms. To do this, we recall that one gram is equal to one thousandth of a kilogram. So multiplying both sides of the equation by 60.5, we find that 60.5 grams is equal to 0.0605 kilograms. And hence, we can change the mass to 0.0605 kilograms, which means we can come back to finding out our change in momentum.
So once again, the change in momentum is equal to the mass of the tennis ball, which we now know to be 0.0605 kilograms, multiplied by the change in velocity of the object, which is 30 meters per second. And this is already in its standard unit.
Now we had kilograms for the mass and meters per second for the velocity. So our change in momentum is going to be in a unit of kilogram meters per second, which once again is the standard unit of momentum or change in momentum, at which point when we evaluate this side of the equation, we find that the change in momentum is equal to 1.815 kilograms meters per second.
At this point, we have a value for the change in momentum, Δ𝑝. And we already knew what 𝐹, the force, was. So we can work out the value of Δ𝑡 by rearranging this equation. We first write down in our table of important information that the change in momentum is 1.815 kilograms meters per second. And then we rearrange to find Δ𝑡 in this equation by dividing both sides of the equation by 𝐹.
Doing this leaves us with Δ𝑡 on the left-hand side. And on the right, we have Δ𝑝 divided by 𝐹, at which point we just need to plug in values. So we have the change in momentum, 1.815 kilograms meters per second, divided by the force, 75 newtons. And once again, because we’re working at standard units on the right-hand side, we’ve got standard units for both change in momentum and force, we’re going to get the time interval in its standard unit of seconds.
So evaluating the right-hand side of the equation, we find that the time interval is equal to 0.0242 seconds. However, this is not our final answer. This is because we need to round our answer to the correct number of significant figures.
Now in the question, we’ve been given one value to three significant figures and two of them to two significant figures, because both 75 and 30 are to two significant figures. Two significant figures, that’s what we need to round our answer to. So here’s the first significant figure and here’s the second.
Now it’s the significant figure afterwards the third significant figure that will tell us what happens to the second significant figure. In this case, the third significant figure is a two. Two is less than five. Therefore, the second significant figure will stay the same. It’s not gonna round up.
Hence, to two significant figures, our time interval is 0.024 seconds. And so we have our final answer. The collision lasts for 0.024 seconds to two significant figures.