A particle of mass and initial velocity strikes a stationary
particle of mass . The collision is elastic. It is observed that after the collision the particles have equal and
opposite velocities: Find (a) the ratio ; (b) the velocity of the CM (before and after the collision); (c)
the total KE of the two particles in the CM frame expressed as a fraction of ; (d) the final KE of in
the CM frame.
(a) Let the masses of the two particles be m1 and m2, respectively. The initial velocity of the first particle is denoted by v1, and the final velocities of both particles after the collision are denoted by v1' and v2', respectively. According to the Law of Conservation of Momentum, the total momentum before the collision is equal to the total momentum after the collision:
m1 * v1 = m1 * v1' + m2 * v2'
Since v1' = -v2' (equal and opposite velocities), we can substitute this into the equation above:
m1 * v1 = m1 * (-v2') + m2 * v2'
Dividing both sides of the equation by m1:
v1 = -v2' + (m2/m1) * v2'
Rearranging the equation to solve for v2':
v2' = (m1 * v1) / (m1 + (m2/m1))
Taking the ratio of v1 to v2' gives us:
(v1 / v2') = (m1 + (m2/m1)) / m1
(b) The velocity of the center of mass (CM) before and after the collision remains unchanged. Therefore, the velocity of the CM is the same before and after the collision.
(c) Now, let's consider the total kinetic energy (KE) of the two particles in the CM frame. In the CM frame, the velocity of the CM is zero. Thus, the total KE of the two particles in the CM frame is equal to the sum of their individual KEs.
KE_total = KE1_CM + KE2_CM
Since the velocity of the CM is zero, the KE in the CM frame for each particle is given by:
KE_CM = (1/2) * m * v^2
Therefore, the total KE of the two particles in the CM frame is:
KE_total = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2
To express this total KE as a fraction of (m1 + m2), we divide both sides of the equation by (m1 + m2):
KE_total / (m1 + m2) = [(1/2) * m1 * v1^2 + (1/2) * m2 * v2^2] / (m1 + m2)
(d) Finally, to find the final KE of m2 in the CM frame, we need to express it as a fraction of (m1 + m2). Since the total KE of the two particles in the CM frame is already expressed as a fraction of (m1 + m2) in part (c), the final KE of m2 in the CM frame can be found by subtracting the KE_CM of m1 from the total KE.
Final KE of m2 in the CM frame = KE_total / (m1 + m2) - KE1_CM
To find the solutions to the given problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
Let's break down each part of the problem step by step:
(a) To find the ratio of the masses, let's denote the mass of the first particle as m1 and the mass of the second particle as m2. The ratio of their masses is given as:
m1/m2 = ?
Since no value is provided for the masses m1 and m2, we cannot calculate the ratio. Please provide the specific values for m1 and m2 to proceed further.
(b) To find the velocity of the center of mass (CM) before and after the collision, we can use the equation:
Velocity of CM = (m1 * v1 + m2 * v2) / (m1 + m2)
Where v1 is the initial velocity of the first particle and v2 is the initial velocity of the second particle. Since one particle is stationary and the other has an initial velocity, we can write the equation as:
Velocity of CM = (m1 * v1) / (m1 + m2)
For the velocity after the collision, since the particles have equal and opposite velocities, the equation becomes:
Velocity of CM (after collision) = (-m2 * v2) / (m1 + m2)
(c) To find the total kinetic energy of the two particles in the CM frame expressed as a fraction of the initial kinetic energy, we need to use the concept of the relative velocity of the particles before and after the collision.
Total Kinetic Energy in the CM frame (before collision) = (1/2) * (m1 + m2) * Velocity of CM^2
Total Kinetic Energy in the CM frame (after collision) = (1/2) * (m1 + m2) * Velocity of CM (after collision)^2
To express the ratio of the total kinetic energy in the CM frame before and after the collision, divide the latter equation by the former:
Total KE (after collision) / Total KE (before collision) = [(1/2) * (m1 + m2) * Velocity of CM (after collision)^2] / [(1/2) * (m1 + m2) * Velocity of CM^2]
= (Velocity of CM (after collision) / Velocity of CM)^2
(d) Finally, to find the final kinetic energy of particle m1 in the CM frame, we can use the equation:
Final KE (m1 in CM frame) = (1/2) * m1 * (Velocity of CM (after collision))^2
Please provide the specific values for the masses and initial velocities of the particles to obtain numerical answers.