A ball of mass three kilograms moving in a straight line at 32 centimeters per second collided with another ball of mass 2.5 kilograms that was at rest. Given that the two balls coalesced into one body, determine the speed of this new body.
We’re told that the first ball has a mass of three kilograms. We can call that mass 𝑚 sub one. The initial speed of that ball, given as 32 centimeters per second, we can call 𝑣 sub one. This first ball collides with a second ball of mass 2.5 kilograms. We’ll call that value 𝑚 sub two. Knowing that, after the collision, the two balls coalesce into one body, we want to solve for the speed of that combined body. We’ll call that speed 𝑣 sub 𝑓.
To start on our solution, let’s draw a sketch of this scenario. In this situation, we have 𝑚 sub one and 𝑚 sub two, where 𝑚 sub two initially is at rest and 𝑚 sub one is approaching 𝑚 sub two with speed 𝑣 sub one.
When the two masses collide, they somehow combine or coalesce together into one mass, which then moves with a final speed we’ve called 𝑣 sub 𝑓 that we want to solve for. To solve for that final speed, we can recall the principle of the conservation of momentum.
One way to write this principle mathematically is to say that the initial momentum in a system, 𝑝 sub 𝑖, is equal to its final momentum. Or put another way, the system’s initial mass times its initial speed is equal to its final mass times its final speed.
If we were to write out an expression for the initial momentum of our system, we might first notice that our system has two masses, 𝑚 sub one and 𝑚 sub two, and that each of those masses has an initial speed. 𝑚 sub one has initial speed 𝑣 sub one. And 𝑚 sub two has initial speed of zero. So that second term goes to zero. This means the initial momentum of our system is 𝑚 sub one times 𝑣 sub one.
When we consider the final momentum of our system, we see that, after the collision, these two spheres have combined into one mass whose total mass is equal to the sum of 𝑚 sub one and 𝑚 sub two.
This sphere moves along with some speed, 𝑣 sub 𝑓. So 𝑣 sub 𝑓 times the sum of the masses is the final momentum of our system. By the principle of momentum conservation, we can write that 𝑚 sub one 𝑣 sub one is equal to 𝑚 sub one plus 𝑚 sub two quantity times 𝑣 sub 𝑓.
Rearranging to solve for 𝑣 sub 𝑓, we find it’s equal to the initial momentum divided by the sum of our masses. Since we’re given 𝑚 sub one, 𝑚 sub two, and 𝑣 sub one in our problem statement, we’re ready to plug in and solve for 𝑣 sub 𝑓.
When we enter these values on our calculator, we find that 𝑣 sub 𝑓 is equal to 17.45 centimeters per second. That’s the final speed of our system of the two combined masses.