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A 2kg ball moving at 24 ms-1 undergoes inelastic head-on collision with a 4 kg ball moving in the opposite direction at 48 ms-1. If the coefficient of restitution is 2/3, their velocities in ms-1 after impact are
- a)-56,-8
- b)-28,-4
- c)-14,-2
- d)-7,-1
Correct answer is option 'A'. Can you explain this answer?
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A 2kg ball moving at 24 ms-1 undergoes inelastic head-on collision wit...
Let's solve the problem step by step:
Given:
Mass of the first ball (m1) = 2 kg
Initial velocity of the first ball (u1) = 24 m/s
Mass of the second ball (m2) = 4 kg
Initial velocity of the second ball (u2) = -48 m/s (opposite direction)
Coefficient of restitution (e) = 2/3
Step 1: Conservation of Momentum
In an inelastic collision, the total momentum before the collision is equal to the total momentum after the collision. We can use this principle to set up an equation:
Initial momentum = Final momentum
(m1 * u1) + (m2 * u2) = (m1 * v1) + (m2 * v2)
where v1 and v2 are the final velocities of the first and second ball, respectively.
Substituting the given values:
(2 kg * 24 m/s) + (4 kg * -48 m/s) = (2 kg * v1) + (4 kg * v2)
48 kg m/s - 192 kg m/s = 2 kg * v1 + 4 kg * v2
-144 kg m/s = 2 kg * v1 + 4 kg * v2
Step 2: Coefficient of Restitution
The coefficient of restitution (e) is defined as the ratio of the relative velocity after the collision to the relative velocity before the collision. Mathematically, it can be expressed as:
e = (v2 - v1) / (u1 - u2)
Substituting the given values:
2/3 = (v2 - v1) / (24 m/s + 48 m/s)
2/3 = (v2 - v1) / 72 m/s
v2 - v1 = (2/3) * 72 m/s
v2 - v1 = 48 m/s
Step 3: Solving the Equations
We now have two equations:
-144 kg m/s = 2 kg * v1 + 4 kg * v2
v2 - v1 = 48 m/s
We can solve these equations simultaneously to find the values of v1 and v2.
Multiplying the second equation by 2:
2(v2 - v1) = 2 * 48 m/s
2v2 - 2v1 = 96 m/s
Adding this equation to the first equation:
-144 kg m/s + 2v2 - 2v1 = 2 kg * v1 + 4 kg * v2
Rearranging the equation:
-2v1 - 4v2 + 2v2 + 2v1 = -144 kg m/s + 96 m/s
-2v2 = -48 kg m/s
Dividing by -2:
v2 = 24 m/s
Substituting the value of v2 in the second equation:
24 m/s - v1 = 48 m/s
Rearranging the equation:
-v1 = 48 m/s - 24 m/s
-v1 = 24 m/s
Dividing by -1:
v1 = -24 m/s
Therefore, the velocities of the balls after the impact are v1 = -24 m/s and v2
Given:
Mass of the first ball (m1) = 2 kg
Initial velocity of the first ball (u1) = 24 m/s
Mass of the second ball (m2) = 4 kg
Initial velocity of the second ball (u2) = -48 m/s (opposite direction)
Coefficient of restitution (e) = 2/3
Step 1: Conservation of Momentum
In an inelastic collision, the total momentum before the collision is equal to the total momentum after the collision. We can use this principle to set up an equation:
Initial momentum = Final momentum
(m1 * u1) + (m2 * u2) = (m1 * v1) + (m2 * v2)
where v1 and v2 are the final velocities of the first and second ball, respectively.
Substituting the given values:
(2 kg * 24 m/s) + (4 kg * -48 m/s) = (2 kg * v1) + (4 kg * v2)
48 kg m/s - 192 kg m/s = 2 kg * v1 + 4 kg * v2
-144 kg m/s = 2 kg * v1 + 4 kg * v2
Step 2: Coefficient of Restitution
The coefficient of restitution (e) is defined as the ratio of the relative velocity after the collision to the relative velocity before the collision. Mathematically, it can be expressed as:
e = (v2 - v1) / (u1 - u2)
Substituting the given values:
2/3 = (v2 - v1) / (24 m/s + 48 m/s)
2/3 = (v2 - v1) / 72 m/s
v2 - v1 = (2/3) * 72 m/s
v2 - v1 = 48 m/s
Step 3: Solving the Equations
We now have two equations:
-144 kg m/s = 2 kg * v1 + 4 kg * v2
v2 - v1 = 48 m/s
We can solve these equations simultaneously to find the values of v1 and v2.
Multiplying the second equation by 2:
2(v2 - v1) = 2 * 48 m/s
2v2 - 2v1 = 96 m/s
Adding this equation to the first equation:
-144 kg m/s + 2v2 - 2v1 = 2 kg * v1 + 4 kg * v2
Rearranging the equation:
-2v1 - 4v2 + 2v2 + 2v1 = -144 kg m/s + 96 m/s
-2v2 = -48 kg m/s
Dividing by -2:
v2 = 24 m/s
Substituting the value of v2 in the second equation:
24 m/s - v1 = 48 m/s
Rearranging the equation:
-v1 = 48 m/s - 24 m/s
-v1 = 24 m/s
Dividing by -1:
v1 = -24 m/s
Therefore, the velocities of the balls after the impact are v1 = -24 m/s and v2
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A 2kg ball moving at 24 ms-1 undergoes inelastic head-on collision with a 4 kg ball moving in the opposite direction at 48 ms-1. If the coefficient of restitution is 2/3, their velocities in ms-1 after impact area)-56,-8b)-28,-4c)-14,-2d)-7,-1Correct answer is option 'A'. Can you explain this answer?
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