Two balls of equal mass, moving with speeds of 3 m/s, collide head-on. Find the speed of each after impact if (a) they stick together, (b) the collision is perfectly elastic, (c) the coefficient of restitution is 1/3.
(a)
The speed of two balls of equal masses after collision if they stick together, given that both the balls move with the speed of
Answer to Problem 47SP
Solution:
Explanation of Solution
Given data:
Both the balls have equal masses.
The balls collide head on.
Both the balls before collision were moving at a speed of
Formula used:
The momentum of a body is expressed as,
Here,
Understand that during a collision between bodies A and B, when the bodies after the collision combine to form a single body, the momentum before and after the collision is expressed as,
Here,
Explanation:
Understand that both the balls are moving with the same speed and they collide head on. Therefore, both must be moving toward each other with velocities of equal magnitude from opposite directions. Therefore, the velocity of one ball is negative of the other. Consider the ball moving with positive velocity as body A and the ball moving with negative velocity as body B.
Consider the expression for initial momentum of body A before collision in horizontal direction
Here,
Substitute
Consider the expression for initial momentum of body B before collision
Here,
Understand that body B has negative velocity. Therefore, substitute
Consider the mass of the final combined body after the collision
Here,
Understand that both the balls stick together after the collision. Consider the expression for the momentum of the combined system after collision
Here,
Substitute
Consider the expression for conservation of momentum for the collision
Substitute
The final speed of the combined system is
Conclusion:
Therefore, the speed of both the balls after impact is
(b)
The speed of two balls of equal masses after the perfectly elastic collision, given that both the balls move with the speed of
Answer to Problem 47SP
Solution: Both rebound at
Explanation of Solution
Given data:
Both the balls have equal masses.
The balls collide head on.
Both the balls before collision were moving at a speed of
Collision is elastic.
Formula used:
The momentum of a body is expressed as,
Here,
The expression for kinetic energy of the body is written as,
Here,
Understand that during a collision between bodies A and B, the conservation of momentum for the collision is expressed as,
Here,
The expression for conservation of kinetic energy for the perfectlyelastic collision of bodies A and B is written as,
Here,
Explanation:
Understand that both the balls are moving with the same speed and they collide head on. Therefore, both must be moving towards each other with velocities of equal magnitude from opposite directions. Therefore, the velocity of one ball is negative of the other. Consider the ball moving with positive velocity as body A and the ball moving with negative velocity as body B.
Consider the expression for final momentum of body A after collision
Here,
Consider the expression for the momentum of body B after collision
Here,
Consider the expression for conservation of momentum for the collision
Substitute
Further, solve as,
Understand that the collision is elastic. Therefore, consider the expression for kinetic energy of body A before collision
Substitute
Consider the expression for kinetic energy of body B before collision
Substitute
Consider the expression for kinetic energy of body A after collision
Consider the expression for kinetic energy of body B after collision
Using equation 1, substitute
Understand that the kinetic energy of the system is conserved during an elastic collision. Therefore, write the expression for conservation of kinetic energy for the elastic collision of bodies A and B.
Substitute
Further, solve as,
Considering negative sign of the velocity since on a head-on collision, if two objects of equal masses collide with equal velocity then they will rebound. Hence,
Consider the expression for final velocity of body B after collision
Substitute
Conclusion:
Therefore, the speed of both body A and body B, that is the speed of both the balls after the collision impact, is
(c)
The speed of two balls of equal masses after collision if the coefficient of restitution for the collision is
Answer to Problem 47SP
Solution: Both balls rebounds at
Explanation of Solution
Given data:
Both the balls have equal masses.
The balls collide head on.
Both the balls before collision were moving at a speed of
Coefficient of restitution is
Formula used:
The momentum of a body is expressed as,
Here,
Understand that during a collision between bodies A and B, the conservation of momentum for the collision is expressed as,
Here,
The coefficient of restitution for a collision of bodies A and B is expressed as,
Here,
Explanation:
Understand that both the balls are moving with the same speed and they collide head on. Therefore, both must be moving towards each other with velocities of equal magnitude from opposite directions. Therefore, velocity of one ball is negative of the other. Consider the ball moving with positive velocity as body A and the ball moving with negative velocity as body B.
Consider the ball with positive velocity as body A and the ball with negative velocity as body B.
Consider the expression for final momentum of body A after collision.
Consider the expression for the momentum of body B after collision.
Consider the expression for conservation of momentum for the collision.
Substitute
Further, solve as,
Consider the expression for coefficient of restitution for the collision.
Substitute
Further, solve as,
Consider the expression for final velocity of body A after collision
Substitute
The negative sign indicates that body A moves in opposite direction to body B.
Conclusion:
Therefore, the speed of both body A and body B, that is the speed of both the balls after the collision impact, is
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Chapter 8 Solutions
Schaum's Outline of College Physics, Twelfth Edition (Schaum's Outlines)
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