A 0.200 kg croquet ball makes an elastic head-on collision with a second ball initially at rest. The second ball moves off with half the original speed of the first ball.
(a) What is the mass of the second ball?
(b) What fraction of the original kinetic energy (KE/KE) gets transferred to the second ball?
T0 solve this, write the conservation of momentum equations, and the conservation of energy. You have two equations, two unknowns.
To solve this problem, we can use the principles of conservation of momentum and conservation of energy. Let's go step by step to find the answers to the questions.
(a) To find the mass of the second ball, let's denote the mass of the first ball as m1 and the mass of the second ball as m2.
According to the conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision. Mathematically, this can be expressed as:
m1 * v1 = m2 * v2
Where,
m1 = mass of the first ball (given as 0.200 kg)
v1 = initial velocity of the first ball (unknown)
m2 = mass of the second ball (to be found)
v2 = final velocity of the second ball (known as half the original speed of the first ball)
Now, we know that the second ball moves off with half the original speed of the first ball. This means:
v1 / 2 = v2
Substituting the value of v2 in terms of v1 in the momentum equation, we get:
m1 * v1 = m2 * (v1 / 2)
Simplifying this equation, we find:
2 * m1 = m2
Substituting the given value of m1 (0.200 kg), we can calculate the mass of the second ball (m2):
2 * 0.200 kg = m2
m2 = 0.400 kg
Therefore, the mass of the second ball is 0.400 kg.
(b) To find the fraction of the original kinetic energy transferred to the second ball, we need to use the conservation of energy.
The initial kinetic energy (KE_initial) of the system is given by:
KE_initial = (1/2) * m1 * (v1)^2
The final kinetic energy (KE_final) of the system can be expressed as the sum of the kinetic energies of the two balls:
KE_final = (1/2) * m1 * (v2)^2 + (1/2) * m2 * (v2)^2
Substitute the value of v2 in terms of v1 (v1/2) in the equation above, we get:
KE_final = (1/2) * m1 * ((v1/2)^2) + (1/2) * m2 * ((v1/2)^2)
Simplifying this equation, we find:
KE_final = ((1/2) * m1 + (1/8) * m2) * (v1)^2
To find the fraction of the original kinetic energy transferred, we need to calculate the ratio of final kinetic energy (KE_final) to the initial kinetic energy (KE_initial):
Fraction transferred = KE_final / KE_initial
Substituting the values into this equation:
Fraction transferred = (((1/2) * m1 + (1/8) * m2) * (v1)^2) / ((1/2) * m1 * (v1)^2)
Now, substitute the known values for m1, m2, and simplify the equation to find the fraction of the original kinetic energy transferred.
Note: Fraction transferred will be a numerical value less than 1, indicating the fraction or percentage of kinetic energy transferred to the second ball.