A ball of mass m makes a head-on elastic collision with a second ball (at rest) and rebounds with a speed equal to one-third its original speed. What is the mass of the second ball?

7-9

0.5m

Since the collision is elastic then kinetic energy will be constant......1/2m1v1^2=1/2m1v1^2+1/m2v2^2.....

Given m1=m
By combining the equations and making v1 the subject
V1=[{v1(m1-m2)}+(2m2*v2)]/(m1+m2)

Note:v2=0 after computing..the mass of the second ball is 1/2m

To solve this problem, we can apply the principles of conservation of momentum and conservation of kinetic energy.

Let's denote the mass of the first ball as m1 and the mass of the second ball as m2. We are given that the first ball rebounds with a speed equal to one-third of its original speed. So, the final velocity of the first ball, v1f, is equal to one-third of its initial velocity, v1i.

Using the principle of conservation of momentum, we know that the total momentum before the collision is equal to the total momentum after the collision. Since the second ball is at rest, the initial momentum of the system is just the momentum of the first ball:

m1 * v1i = m1 * v1f + m2 * v2f

Now, let's use the principle of conservation of kinetic energy. In an elastic collision, kinetic energy is conserved. The initial kinetic energy of the system is given by:

(1/2) * m1 * v1i^2

And the final kinetic energy is given by:

(1/2) * m1 * v1f^2 + (1/2) * m2 * v2f^2

Since the initial and final velocities of the first ball are related by v1f = (1/3) * v1i, we can rewrite the final kinetic energy as:

(1/18) * m1 * v1i^2 + (1/2) * m2 * v2f^2

Since kinetic energy is conserved, we equate the initial and final kinetic energies:

(1/2) * m1 * v1i^2 = (1/18) * m1 * v1i^2 + (1/2) * m2 * v2f^2

Simplifying this equation, we get:

(17/18) * m1 * v1i^2 = (1/2) * m2 * v2f^2

Now, substituting v1f = (1/3) * v1i, we have:

(17/18) * m1 * v1i^2 = (1/2) * m2 * (v1i/3)^2

Simplifying further, we get:

(17/18) * m1 = (1/18) * m2 / 9

Cross-multiplying and rearranging:

17 * 9 * m1 = m2

Finally, the mass of the second ball, m2, is given by:

m2 = 17 * 9 * m1

Therefore, the mass of the second ball is 153 times the mass of the first ball.