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.