# Question Video: Finding the Impulse Exerted on a Sphere Colliding with Another Sphere Moving in the Opposite Direction Mathematics

Two spheres of masses 200 g and 350 g were moving toward each other along the same horizontal straight line. The first was moving at 14 m/s and the second at 3 m/s. The two spheres collided. As a result, the first sphere rebounded at 7 m/s in the opposite direction. Given that the positive direction is the direction of motion of the first sphere before the impact, determine the impulse 𝐼 the second sphere exerted on the first one and the speed 𝑣 of the second sphere after impact.

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### Video Transcript

Two spheres of masses 200 grams and 350 grams were moving toward each other along the same horizontal straight line. The first was moving at 14 meters per second and the second at three meters per second. The two spheres collided. As a result, the first sphere rebounded at seven meters per second in the opposite direction. Given that the positive direction is the direction of motion of the first sphere before the impact, determine the impulse 𝐼 the second sphere exerted on the first one and the speed 𝑣 of the second sphere after impact.

Now before we go any further, we might observe that with our speeds, we’re working in meters per second, whilst with the masses, we’re working in grams. So we’re going to convert their masses to SI base units of mass, the kilogram. To do so, we divide each by 1000. We get 0.2 kilograms and 0.35 kilograms, respectively.

So how do we calculate the impulse 𝐼 that the second sphere exerted on the first? Well, first, we know that the magnitudes of the impulses on the spheres are equal. We also have information about the velocity of the first sphere before and after the collision. This means we can calculate the change in velocity of the first sphere due to the collision. If we call that the change in 𝑣 sub 𝑎, where 𝑎 is the name of the first sphere, then we can say that this is equal to its final velocity minus its initial velocity.

Now, since it was initially moving in the positive direction, it will rebound and move in the negative direction. This means if it’s for traveling at a speed of seven meters per second, its velocity is negative seven. So the change in velocity is negative seven minus 14. And that’s negative 21 meters per second. Then, we can calculate the impulse on the first sphere by finding its change in momentum. Momentum is of course mass times velocity. So we could calculate the momentum before and after and subtract them. Or alternatively, we can multiply the momentum by the change in velocity. For our first sphere, we can then say that its mass is 0.2 and its change in velocity is negative 21. So the impulse on this sphere is 0.2 times negative 21, and that’s negative 4.2. And the units we use are newton seconds.

Next, we know that an impulse of equal magnitude, but opposite in sign, acts on the second sphere. In other words, the impulse on the second sphere must be 4.2 newton seconds. But of course, we can link that to its change in momentum. This means it’s equal to its mass times its change in velocity. Now, since it was initially moving in the negative direction at a speed of three meters per second, its change in velocity is 𝑣 minus negative three. So its change in momentum is 0.35 times 𝑣 minus negative three. This right-hand side is equivalent to 0.35 times 𝑣 plus three. And we calculated the impulse on the second sphere was 4.2. So we can say 4.2 equals 0.35 times 𝑣 plus three.

Next, we divide through by 0.35. Now, 4.2 divided by 0.35 is simply 12. Then, we subtract three from both sides. 12 minus three is nine. So we’ve calculated that 𝑣 is equal to nine meters per second. And so we’ve answered the question. The impulse that the second sphere exerted on the first, which we defined earlier to be 𝐽 sub 𝑎, is in fact 𝐼 equals negative 4.2 newton seconds. Similarly, the speed 𝑣 of the second sphere is the magnitude of nine meters per second, which is simply nine meters per second.