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A ball is moving in a straight line collide elastically with another ball of same mass such that at the moment of impact the angle Between the line passing through the ball centre at 45 degree with the direction of the initial motion of striking ball if balls are smooth find the maximum fraction of kinetic energy that transfer to potential energy in the process of collision (A)1/4 (B)1/9 (C)1/3 (D)1/2?
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A ball is moving in a straight line collide elastically with another b...
If (v1x,v1y) are the instantaneous velocity components of the incident ball and are the velocity components of the struck ball at the same moment, then since there are no external impulsive forces (i.e. other than the mutual interaction of the balls)
(F is along the x-axis as the balls are smooth. Thus Y component of momentum is not transferred.) Since loss of K.E. is stored as deformation energy D, we have
(F is along the x-axis as the balls are smooth. Thus Y component of momentum is not transferred.) Since loss of K.E. is stored as deformation energy D, we have
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A ball is moving in a straight line collide elastically with another b...
Given information:
- Two balls of the same mass collide elastically.
- The line passing through the center of the striking ball makes a 45-degree angle with the initial motion direction of the striking ball.
- The balls are smooth.
To find: The maximum fraction of kinetic energy transferred to potential energy in the collision.
Solution:
1. Initial conditions:
Let's consider the initial conditions before the collision.
- Ball 1 (striking ball) is moving with an initial velocity V1.
- Ball 2 (struck ball) is initially at rest.
2. Collision:
During the collision, the balls collide elastically, which means kinetic energy is conserved. The angle between the line passing through the center of the striking ball and the direction of its initial motion is 45 degrees.
3. Final conditions:
After the collision, let's analyze the final conditions.
- Ball 1 (striking ball) will rebound in a new direction with a velocity V1'.
- Ball 2 (struck ball) will start moving in the direction of the striking ball's initial motion with a velocity V2.
4. Conservation of momentum:
Since the collision is elastic and the balls have the same mass, the momentum is conserved in both the x and y directions.
- In the x-direction: m1 * V1 = m1 * V1' + m2 * V2 (1)
- In the y-direction: 0 = m1 * V1'*sin(45) + m2 * V2*sin(45) (2)
5. Conservation of kinetic energy:
Since the collision is elastic, the kinetic energy is conserved.
- Initial kinetic energy: (1/2) * m1 * V1^2
- Final kinetic energy: (1/2) * m1 * V1'^2 + (1/2) * m2 * V2^2
6. Finding V1' and V2:
From equations (1) and (2), we can solve for V1' and V2:
- V1' = (m1 - m2) * V1 / (m1 + m2)
- V2 = (2 * m1 * V1) / (m1 + m2)
7. Finding the fraction of kinetic energy transferred to potential energy:
The maximum fraction of kinetic energy transferred to potential energy occurs when the struck ball momentarily stops and changes its direction completely.
- At the moment when the struck ball stops, its final velocity (V2) equals 0.
- Therefore, the final kinetic energy is (1/2) * m1 * V1'^2.
Now, we can calculate the fraction of kinetic energy transferred to potential energy:
- Fraction = (Final kinetic energy) / (Initial kinetic energy)
= (1/2) * m1 * V1'^2 / ((1/2) * m1 * V1^2)
= V1'^2 / V1^2
= [(m1 - m2) * V1 / (m1 + m2)]^2 / V1^2
= [(m1 - m2)^2 * V1^2] / [(m1 +
- Two balls of the same mass collide elastically.
- The line passing through the center of the striking ball makes a 45-degree angle with the initial motion direction of the striking ball.
- The balls are smooth.
To find: The maximum fraction of kinetic energy transferred to potential energy in the collision.
Solution:
1. Initial conditions:
Let's consider the initial conditions before the collision.
- Ball 1 (striking ball) is moving with an initial velocity V1.
- Ball 2 (struck ball) is initially at rest.
2. Collision:
During the collision, the balls collide elastically, which means kinetic energy is conserved. The angle between the line passing through the center of the striking ball and the direction of its initial motion is 45 degrees.
3. Final conditions:
After the collision, let's analyze the final conditions.
- Ball 1 (striking ball) will rebound in a new direction with a velocity V1'.
- Ball 2 (struck ball) will start moving in the direction of the striking ball's initial motion with a velocity V2.
4. Conservation of momentum:
Since the collision is elastic and the balls have the same mass, the momentum is conserved in both the x and y directions.
- In the x-direction: m1 * V1 = m1 * V1' + m2 * V2 (1)
- In the y-direction: 0 = m1 * V1'*sin(45) + m2 * V2*sin(45) (2)
5. Conservation of kinetic energy:
Since the collision is elastic, the kinetic energy is conserved.
- Initial kinetic energy: (1/2) * m1 * V1^2
- Final kinetic energy: (1/2) * m1 * V1'^2 + (1/2) * m2 * V2^2
6. Finding V1' and V2:
From equations (1) and (2), we can solve for V1' and V2:
- V1' = (m1 - m2) * V1 / (m1 + m2)
- V2 = (2 * m1 * V1) / (m1 + m2)
7. Finding the fraction of kinetic energy transferred to potential energy:
The maximum fraction of kinetic energy transferred to potential energy occurs when the struck ball momentarily stops and changes its direction completely.
- At the moment when the struck ball stops, its final velocity (V2) equals 0.
- Therefore, the final kinetic energy is (1/2) * m1 * V1'^2.
Now, we can calculate the fraction of kinetic energy transferred to potential energy:
- Fraction = (Final kinetic energy) / (Initial kinetic energy)
= (1/2) * m1 * V1'^2 / ((1/2) * m1 * V1^2)
= V1'^2 / V1^2
= [(m1 - m2) * V1 / (m1 + m2)]^2 / V1^2
= [(m1 - m2)^2 * V1^2] / [(m1 +
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A ball is moving in a straight line collide elastically with another ball of same mass such that at the moment of impact the angle Between the line passing through the ball centre at 45 degree with the direction of the initial motion of striking ball if balls are smooth find the maximum fraction of kinetic energy that transfer to potential energy in the process of collision (A)1/4 (B)1/9 (C)1/3 (D)1/2?
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A ball is moving in a straight line collide elastically with another ball of same mass such that at the moment of impact the angle Between the line passing through the ball centre at 45 degree with the direction of the initial motion of striking ball if balls are smooth find the maximum fraction of kinetic energy that transfer to potential energy in the process of collision (A)1/4 (B)1/9 (C)1/3 (D)1/2? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A ball is moving in a straight line collide elastically with another ball of same mass such that at the moment of impact the angle Between the line passing through the ball centre at 45 degree with the direction of the initial motion of striking ball if balls are smooth find the maximum fraction of kinetic energy that transfer to potential energy in the process of collision (A)1/4 (B)1/9 (C)1/3 (D)1/2? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A ball is moving in a straight line collide elastically with another ball of same mass such that at the moment of impact the angle Between the line passing through the ball centre at 45 degree with the direction of the initial motion of striking ball if balls are smooth find the maximum fraction of kinetic energy that transfer to potential energy in the process of collision (A)1/4 (B)1/9 (C)1/3 (D)1/2?.
A ball is moving in a straight line collide elastically with another ball of same mass such that at the moment of impact the angle Between the line passing through the ball centre at 45 degree with the direction of the initial motion of striking ball if balls are smooth find the maximum fraction of kinetic energy that transfer to potential energy in the process of collision (A)1/4 (B)1/9 (C)1/3 (D)1/2? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A ball is moving in a straight line collide elastically with another ball of same mass such that at the moment of impact the angle Between the line passing through the ball centre at 45 degree with the direction of the initial motion of striking ball if balls are smooth find the maximum fraction of kinetic energy that transfer to potential energy in the process of collision (A)1/4 (B)1/9 (C)1/3 (D)1/2? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A ball is moving in a straight line collide elastically with another ball of same mass such that at the moment of impact the angle Between the line passing through the ball centre at 45 degree with the direction of the initial motion of striking ball if balls are smooth find the maximum fraction of kinetic energy that transfer to potential energy in the process of collision (A)1/4 (B)1/9 (C)1/3 (D)1/2?.
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