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A sphere A of mass m moving with constant velocity u hits another stationary sphere B of the same mass in a head on collision. if e is the coefficient of restitution, what is the ratio of velocities of sphere A to the sphere B after collision?
- a)(1–e)/(1+e)
- b)(1 + e)/(1–e)
- c)(e + 1)/(e–1)
- d)(e–1)/(e +1)
Correct answer is option 'A'. Can you explain this answer?
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A sphere A of mass m moving with constant velocity u hits another stat...
Solution:
Given data,
Mass of sphere A = Mass of sphere B = m
Initial Velocity of sphere A = u
Initial Velocity of sphere B = 0
Coefficient of restitution = e
Let us consider the collision in the frame of reference of sphere B.
In this frame of reference, sphere B is at rest before and after the collision.
Therefore, the velocity of sphere A before collision = u - 0 = u
Let the velocity of sphere A after collision be v1 and that of sphere B after collision be v2.
Using the conservation of momentum in the direction of collision, we get:
mu = mv1 + mv2
Dividing the equation by m, we get:
u = v1 + v2
Using the coefficient of restitution, we get:
e = (v2 - v1)/(u - 0)
e(u - 0) = v2 - v1
v2 - v1 = eu
Adding v1 on both sides, we get:
v2 = v1 + eu
Substituting the value of v2 in the momentum conservation equation, we get:
mu = mv1 + m(v1 + eu)
mu = mv1 + mv1 + meu
mu = 2mv1 + meu
v1 = (mu)/(2m(1 - e))
v2 = v1 + eu
v2 = [(mu)/(2m(1 - e))] + eu
v2 = [(mu)/(2m(1 - e))] + e(u)
v2 = [u(m(1+e))]/[2m(1-e)]
The ratio of velocities of sphere A to sphere B after collision is:
v1/v2 = [(mu)/(2m(1 - e))] / {[u(m(1+e))]/[2m(1-e)]}
v1/v2 = (1 - e)/(1 + e)
v1/v2 = (1e)/(1+e)
Thus, option A is the correct answer.
Given data,
Mass of sphere A = Mass of sphere B = m
Initial Velocity of sphere A = u
Initial Velocity of sphere B = 0
Coefficient of restitution = e
Let us consider the collision in the frame of reference of sphere B.
In this frame of reference, sphere B is at rest before and after the collision.
Therefore, the velocity of sphere A before collision = u - 0 = u
Let the velocity of sphere A after collision be v1 and that of sphere B after collision be v2.
Using the conservation of momentum in the direction of collision, we get:
mu = mv1 + mv2
Dividing the equation by m, we get:
u = v1 + v2
Using the coefficient of restitution, we get:
e = (v2 - v1)/(u - 0)
e(u - 0) = v2 - v1
v2 - v1 = eu
Adding v1 on both sides, we get:
v2 = v1 + eu
Substituting the value of v2 in the momentum conservation equation, we get:
mu = mv1 + m(v1 + eu)
mu = mv1 + mv1 + meu
mu = 2mv1 + meu
v1 = (mu)/(2m(1 - e))
v2 = v1 + eu
v2 = [(mu)/(2m(1 - e))] + eu
v2 = [(mu)/(2m(1 - e))] + e(u)
v2 = [u(m(1+e))]/[2m(1-e)]
The ratio of velocities of sphere A to sphere B after collision is:
v1/v2 = [(mu)/(2m(1 - e))] / {[u(m(1+e))]/[2m(1-e)]}
v1/v2 = (1 - e)/(1 + e)
v1/v2 = (1e)/(1+e)
Thus, option A is the correct answer.
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A sphere A of mass m moving with constant velocity u hits another stationary sphere B of the same mass in a head on collision. if e is the coefficient of restitution, what is the ratio of velocities of sphere A to the sphere B after collision?a)(1–e)/(1+e)b)(1 + e)/(1–e)c)(e + 1)/(e–1)d)(e–1)/(e +1)Correct answer is option 'A'. Can you explain this answer?
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