A ball of mass 60 grams started accelerating from rest at seven centimetres per square second. At the same moment, another ball of mass 40 grams, which was 450 centimetres ahead of the first ball. Started moving in the same direction of the first ball at a constant speed of 90 centimetres per second. Then the two balls collided and coalesced into one body. Find the speed of this body directly after the collision.
There is rather a lot of information in this question. A key hint as to what we might need to do is the fact that the balls collided and coalesced into one body. And as such, we should be starting to think about the law of conservation of momentum. This tells us that the total momentum before a collision is equal to the total momentum after the collision. Where momentum, 𝑃, is equal to mass times velocity of the object.
We’re now going to draw a sketch. We have our first ball. It has a mass of 60 grams and accelerates from rest. That’s a starting velocity of zero. It accelerates at a rate of seven centimetres per square second. At the same time, another ball, which has a mass of 40 grams, which is 450 centimetres ahead of the first ball, moves in the same direction. But this time at a constant speed of 90 centimetres per second.
At some point, and we don’t actually know when, the two balls collide and they coalesce into one body. The mass of the new body is the sum of 60 grams and 40 grams. It’s 100 grams. And of course, we’re looking to find its speed. So let’s call that 𝑣 centimetres per second. Now we said that the total momentum before is equal to the total momentum after. But what is the total momentum before the collision?
To calculate this, since momentum is mass times velocity, we need to know the velocity of the 60-gram ball as the two balls collide. So we’re going to use the equations of constant acceleration to do so. These are sometimes called the SUVAT equations.
We know that when the two balls collide, their displacement from a fixed point — let’s set that equal to zero here — must be the same. Well, the starting speed of the first ball is zero. So using the equation 𝑠 equals 𝑢𝑡 plus half 𝑎𝑡 squared, we get 𝑠 equals zero plus half times 70 squared, since acceleration is seven. That’s 𝑠 equals 3.5𝑡 squared. But what about the second ball?
Well, we’ve already seen that this ball starts 450 centimetres away from the value we’ve set to be equal to zero. It’s moving at a constant speed. So its acceleration is zero. So if we were just using 𝑠 equals 𝑢𝑡 plus half 𝑎𝑡 squared, we get 𝑠 equals 90𝑡. But this ball will always be 450 centimetres further away from the starting point. So we say 𝑠 is equal to 450 plus 90𝑡.
Now when the two balls collide, the displacement from this starting point will be equal. So we can say that 3.5𝑡 squared equals 450 plus 90𝑡. We rearrange by subtracting 90𝑡 and 450 from both sides. And we get the equation 3.5𝑡 squared minus 90𝑡 minus 450 equals zero.
There are a number of ways we can solve this. We could use the quadratic formula. Whatever method we choose, when we solve for 𝑡, we get 𝑡 equals 30 or negative 30 over seven. But of course 𝑡 is time. So we’re going to disregard this negative value. And we see then that the balls collide 30 seconds after they start moving.
We want to know the velocity of the first ball, the 60-gram ball, at this point. So we use the formula 𝑣 equals 𝑢 plus 𝑎𝑡. 𝑢 is zero, 𝑎 is seven, and we now know 𝑡 is equal to 30. So we find the velocity of the first ball at the point of collision to be 210 centimetres per second.
We can now apply the principle of conservation of momentum. Let’s clear some space. Since the balls collide when the velocity of the first ball is 210 centimetres per second, we can say the momentum of the first ball at this point is 60 times 210. And the momentum of the second ball is 40 times 90.
Now we wouldn’t always work in grams and centimetres per second. But in this case, as long as we’re consistent throughout, it really doesn’t matter. After the collision, the total mass is 100 grams. And we’ve said its velocity or its speed is 𝑣. On the left-hand side, this simplifies to 16200. And we can, of course, solve this equation for 𝑣 by dividing through by 100. Well, 16200 divided by 100 is 162. And we said we need to be consistent with our units. So 𝑣 is equal to 162 centimetres per second.
Now in fact, what we’ve done is calculate the velocity of the object. Of course, speed is simply the magnitude of velocity. So any signs don’t actually matter. And we find the speed to be 162 centimetres per second.