A ball is dropped onto a horizontal floor. It reaches a height of 144 cm on the first bounce, an

A ball is dropped onto a horizontal floor. It reaches a...

A ball is dropped onto a horizontal floor. It reaches a height of 144 $\mathrm{cm}$ on the first bounce, and $81 \mathrm{~cm}$ on the second bounce. Find $(a)$ the coefficient of restitution between the ball and floor and $(b)$ the height it attains on the third bounce.

Key Concepts

Energy Transformation in Bouncing Dynamics

When a ball bounces, its gravitational potential energy is converted into kinetic energy during the fall, and then partially back into potential energy as it rebounds. However, due to energy losses in the collision (characterized by the coefficient of restitution), the rebound height is lower than the previous height. This conversion and loss of energy explains the progressive decrease in bounce heights.

Coefficient of Restitution

This concept represents the ratio of speeds after and before an impact in a collision. In the context of bouncing balls, it can be determined by taking the square root of the ratio of the rebound height to the original drop height. It provides a measure of how 'elastic' the collision is, with a value of 1 indicating a perfectly elastic collision (no energy loss) and values less than 1 indicating energy loss during the impact.

Transcript

00:01 Hi, this question, given that a ball comes from a height of one and bounces to 81 centimeters the ones find the coefficient of restitution.

00:13 This is the formal of the question of the restitution and v2 taking object 1 as the ground and object 2 as the ball so velocity of the ball final velocity of pull has v2, final velocity of the ground v1 initial velocity of the ground is u1 final velocity initial velocity of the ball is 2 v1 and u1 will be zero because the wall is stationary put after and before collision so the formal will become v2 minus u 2 and now the kinetic energy just after collision with the ground will be equal to the potential energy that and that will determine by the height the ball reaches and so kinetic energy is k a is equal to 1 over 2 mv squared.

01:20 Potential energy is m g h.

01:24 So this is after collision.

01:33 Kinetic energy 1 over 2 m e squared equal to potential energy and we can cancel out the masses here.

01:53 So we'll get v is equal to squared.

01:58 And this will be v2 which is the final velocity of the ball after collision and we can also do for b4 collision to determine you so 1 over 2m u 2m u 2 squared is equal to m g h and wherever the same u 2 is equal to 2 g h is equal to the square root of 2 g h and the head shear will be the height of the ball was before collision so the ball came from the height of one and there are 44 centimeters and it's called this h1 h2 so h1 is equal to 1 4 centimeters h2 is equal to 81 centimeters now again so now e.

03:04 Will be equal to minus v2 over u2 which will be squared of 2g over square of 2g h over square of 2g h1 this negative here substituting get is equal to minus squared of so 2g can so is out here squared of h2 over h1 substituting we get minus u21 about 144 b equal to and actually for this as the ball bounces like this and rebound here we can take down as the positive direction so comes with positive view and it goes you two goes with minus v2 so we need to take our science this is going to consideration on when doing this so v2 actually be minus 2 to h2.

04:26 So yeah...

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