A billiards cue ball of mass 0.380 kg has a perfectly elastic head-on collision with a standard pool ball. The second pool ball which started at rest travels with a speed which is half the original speed of the cue ball.
(a) Calculate the mass of the pool ball.
kg
(b) Calculate the fraction of the original kinetic energy (ÄKE/KE) that gets transferred to the pool ball.
To solve this problem, we can apply the law of conservation of momentum and the law of conservation of kinetic energy.
Let's begin by calculating the mass of the pool ball.
(a) Calculate the mass of the pool ball:
The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision, assuming no external forces are acting on the system.
Given:
Mass of the cue ball (m1) = 0.380 kg
Initial speed of the cue ball (u1) = ?
Mass of the pool ball (m2) = ?
Initial speed of the pool ball (u2) = 0 m/s (as it started at rest)
Let's denote the final velocities of the cue ball and pool ball as v1 and v2, respectively.
By applying the conservation of momentum:
Initial momentum = Final momentum
(m1 * u1) + (m2 * u2) = (m1 * v1) + (m2 * v2)
Since the second pool ball is at rest initially (u2 = 0), the equation simplifies to:
m1 * u1 = m1 * v1
Given that the speed of the pool ball is half of the speed of the cue ball (v2 = 0.5 * u1), we can substitute these values into the equation:
m1 * u1 = m1 * v1
0.380 kg * u1 = 0.380 kg * v1
Canceling out the mass term:
u1 = v1
Therefore, the mass of the pool ball is the same as the mass of the cue ball:
m2 = 0.380 kg
So, the mass of the pool ball is 0.380 kg.
(b) Calculate the fraction of the original kinetic energy (ΔKE/KE) that gets transferred to the pool ball:
The law of conservation of kinetic energy states that the total kinetic energy before the collision is equal to the total kinetic energy after the collision, assuming no external forces are acting on the system.
Given:
Initial kinetic energy of the cue ball (KE1) = 0.5 * m1 * (u1)^2
Final kinetic energy of the cue ball (KE1') = 0.5 * m1 * (v1)^2
Initial kinetic energy of the pool ball (KE2) = 0.5 * m2 * (u2)^2
Final kinetic energy of the pool ball (KE2') = 0.5 * m2 * (v2)^2
To calculate the fraction of the original kinetic energy transferred to the pool ball, we need to find the change in kinetic energy (ΔKE) and divide it by the initial kinetic energy (KE1).
ΔKE = KE1' - KE1
= (0.5 * m1 * (v1)^2) - (0.5 * m1 * (u1)^2)
= 0.5 * m1 * ((v1)^2 - (u1)^2)
So, the fraction of the original kinetic energy transferred to the pool ball is:
(ΔKE / KE1) = (0.5 * m1 * ((v1)^2 - (u1)^2)) / (0.5 * m1 * (u1)^2)
Canceling out the mass term and simplifying the equation, we have:
(ΔKE / KE1) = ((v1)^2 - (u1)^2) / (u1)^2
Since we know u1 = v1, we can substitute this value into the equation:
(ΔKE / KE1) = ((v1)^2 - (u1)^2) / (u1)^2
= (v1^2 - u1^2) / u1^2
Since v1 = 0.5 * u1, we can further substitute this value:
(ΔKE / KE1) = ((0.5 * u1)^2 - u1^2) / (u1)^2
Simplifying the equation:
(ΔKE / KE1) = (0.5^2 * u1^2 - u1^2) / u1^2
= (0.25 - 1) / 1
= -0.75 / 1
= -0.75
Therefore, the fraction of the original kinetic energy transferred to the pool ball is -0.75 or -75%.
Note: The negative sign indicates that the kinetic energy of the system has decreased after the collision.