Two spheres, A and B, are moving in a straight line on a smooth horizontal plane in opposite directions at 7.17 metres per second. If their masses are four m and eight m, respectively, find the velocity 𝑣 AB of sphere A relative to sphere B. Given that the two bodies coalesce on impact into one body, find the speed 𝑣 of this new body just after the collision.
So the first thing we’re asked to do here is find the velocity 𝑣 AB of sphere A relative to sphere B. Let’s draw a little sketch and see if we can work out what’s happening. We have sphere A with the mass of four m moving in one direction at 7.17 metres per second and a mass B moving in the opposite direction at the same speed. 𝑣 AB is the relative velocity of sphere A to sphere B. Now, sphere B is moving at 7.17 metres per second in one direction. We’re going to subtract the velocity of sphere A. Now, that’s negative 7.17 because that’s moving in the other direction. 7.17 minus negative 7.17 is 7.17 plus 7.17. That’s 14.34 and the units remain the same. So 𝑣 AB is 14.34 metres per second.
Now, the next part of this question asks us to find the speed of the new body, that’s once the bodies coalesce, just after the collision. We might assume, since the mass of B is greater than the mass of A, that the velocity is acting in the same direction as the velocity for B after the collision. The total mass of the new body is four m plus eight m. So it’s 12 m. And now, we apply the law of conservation of momentum. This says that the total momentum before the collision must be equal to the total momentum after. If we let the new body be C, we can say that the momentum of A plus the momentum of B must be equal to the momentum of C. Where 𝑃, which is momentum, is equal to mass times velocity.
We begin by calculating the momentum of sphere A. It’s four m times negative 7.17. And the momentum of B is eight m times 7.17. Now, of course, we could have assumed that the other direction was positive. It really doesn’t matter as long as we’re consistent throughout the question. The momentum of sphere C is 12 m times 𝑣. We divide through by m. And this simplifies to 28.68 equals 12𝑣. We solved this equation for 𝑣 by dividing through by 12. And we get 2.39. Again, the units remain unchanged. So the speed is 2.39 metres per second.
Now, had we, in fact, chosen the other direction to be positive, we would’ve ended up with a velocity of negative 2.39 metres per second. Of course, though, we’re interested in the speed which is the magnitude of velocity. So the sign doesn’t matter. The answer is 2.39 metres per second.