# Question Video: Finding the Velocity of a Sphere after Collision with an Identical Sphere at the Same Line Mathematics

Two spheres 𝐴 and 𝐵, of equal mass, were projected toward each other along a horizontal straight line at 19 cm/s and 29 cm/s respectively. As a result of the impact, sphere 𝐵 rebounded at 10 cm/s. Find the velocity of sphere 𝐴 after the collision given that its initial direction is the positive direction.

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### Video Transcript

Two spheres 𝐴 and 𝐵 of equal mass were projected toward each other along a horizontal straight line at 19 centimeters per second and 29 centimeters per second, respectively. As a result of the impact, sphere 𝐵 rebounded at 10 centimeters per second. Find the velocity of sphere 𝐴 after the collision given that its initial direction is the positive direction.

In order to answer this question, we will draw two diagrams modeling the situation before and after the collision. Before the collision, the two spheres of equal mass 𝑚 are moving toward each other, sphere 𝐴 at 19 centimeters per second and sphere 𝐵 at 29 centimeters per second. We are told that after the collision, sphere 𝐵 rebounds at 10 centimeters per second. And we need to calculate the velocity of sphere 𝐴. We’re also told that sphere 𝐴 was initially moving in the positive direction. In order to answer this question, we will use the conservation of momentum. This states that the momentum before is equal to the momentum after.

As momentum is equal to mass multiplied by velocity, our equation is 𝑚 one 𝑢 one plus 𝑚 two 𝑢 two is equal to 𝑚 one 𝑣 one plus 𝑚 two 𝑣 two. Substituting in the initial velocities, we have 𝑚 multiplied by 19 plus 𝑚 multiplied by negative 29. This is because sphere 𝐵 is moving in the negative direction. This is equal to 𝑚 multiplied by 𝑣, the velocity of sphere 𝐴 after the collision, plus 𝑚 multiplied by 10.

As the mass of both spheres is the same and cannot be equal to zero, we can divide through by the variable 𝑚. 19 plus negative 29 is equal to negative 10. This leaves us with negative 10 is equal to 𝑣 plus 10. We can then subtract 10 from both sides of this equation giving us 𝑣 is equal to negative 20. The velocity of sphere 𝐴 after the collision is negative 20 centimeters per second. This means that it is moving with a speed of 20 centimeters per second in the negative direction.

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