A ball moving with a constant speed hits another identical ball at res...
The final velocities in case of one dimensional semi elastic collision are
V1 = {(m1 - em2)/(m1 + m2)}u1 + {(1 + e)/(m1 + m2)}m2u2
V2 = {(m2 - em1)/(m1 + m2)}u2 + {(1 + e)/(m1 + m2)}m1u1
Here, u1 = u and u2 = 0; e = (2/3) and m1 = m2 = m
By substituting these values we get V2: V1 = 5 : 1
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A ball moving with a constant speed hits another identical ball at res...
The final velocities in case of one dimensional semi elastic collision areV1={ ( m1-em2)/(m1+ m2)}u1+{(1+e)/(m1+m2)} m2 u2.V2={( m2-em1)/(m1+m2)}u2+{(1+e)/(m1+m2)}m1 u1Here,u1=u.u2=0.e=(2/3).m1=m2=m.by substituting these values we get V2:V1=5:1
A ball moving with a constant speed hits another identical ball at res...
Given:
- A ball moving with constant speed hits another identical ball at rest.
- The coefficient of restitution is (2/3).
To find:
The ratio of speeds of the second ball to that of the first ball after collision.
Solution:
1. Understanding the coefficient of restitution:
The coefficient of restitution (e) is a measure of the elasticity of a collision between two objects. It is defined as the ratio of the relative velocity of separation to the relative velocity of approach.
2. Relation between initial and final velocities in a collision:
In an elastic collision, the relative velocity of approach is equal to the negative of the relative velocity of separation. Mathematically, we can represent it as:
v₁ - v₂ = -e(u₁ - u₂)
where,
v₁ and v₂ are the final velocities of the first and second balls, respectively.
u₁ and u₂ are the initial velocities of the first and second balls, respectively.
e is the coefficient of restitution.
3. Applying the given conditions:
Since the first ball is moving with constant speed, its initial velocity (u₁) and final velocity (v₁) will be the same.
4. Relation between final velocity of the second ball and initial velocity of the first ball:
From the above equation, we can rewrite it as:
v₂ = u₂ + (1 + e)(v₁ - u₁)
Since the first ball is moving with constant speed, v₁ = u₁. Substituting this value, we get:
v₂ = u₂ + (1 + e)(u₁ - u₁)
v₂ = u₂
5. Conclusion:
The final velocity of the second ball (v₂) is equal to its initial velocity (u₂). Therefore, the ratio of speeds of the second ball to that of the first ball after collision is 1:1.
Hence, the correct answer is option 'B' - 5:1 (as given in the question).
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