It was observed that a tennis ball when served horizontally...

It was observed that a tennis ball when served horizontally 7.5 ft above the ground strikes the smooth ground at $B 20 \mathrm{ft}$ away. Determine the initial velocity $v_{A}$ of the ball and the velocity $v_{B}(\text { and } \theta)$ of the ball just after it strikes the court at $B$. Take $e=0.7$.

Key Concepts

Conservation of Momentum

This principle states that in a closed system with no external forces the total momentum remains constant before and after a collision. It is fundamental for analyzing collision problems, as it provides an equation that relates the masses and velocities of objects involved in the collision, allowing one to solve for unknown final velocities.

Coefficient of Restitution

The coefficient of restitution is a measure of the elasticity of a collision, defined as the ratio of the relative speed after collision to the relative speed before collision along the line of impact. It quantifies how much kinetic energy is conserved in a collision, with a value of 1 representing a perfectly elastic collision and lower values indicating inelastic behavior.

Sequential Collision Analysis

Sequential collision analysis involves examining collisions that occur one after the other, where the outcome of one collision serves as the initial condition for the next. It requires applying conservation of momentum and the coefficient of restitution for each individual collision in a step-by-step manner to determine the overall system behavior.

Transcript

00:02 In this problem, we have a tennis ball which is served horizontally at a given height above the ground and then strikes the ground a distance 20 feet away.

00:13 We want to calculate the initial velocity of the ball and the velocity and direction of the ball after it strikes the ground.

00:21 Using kinematic first and taking the right direction as positive, we'll use the equation s is equal to s -0 plus v -0 -t.

00:32 So this is in the horizontal direction.

00:35 The ball covers a distance of 20 feet from a starting position of 0, the initial velocity va times time t.

00:46 Both of these are unknown.

00:49 We'll then apply the same equation to the direction in the vertical, to the motion in the vertical direction, and we'll take down as positive.

01:01 So here s is equal to s -0 plus v -0, t, plus a half a t squared.

01:13 So the vertical distance is 7 .5 feet.

01:19 The ball starts at position 0 and v.

01:24 T is also 0 since the ball is officially addressed.

01:27 So this is equal to a half and the acceleration is due to gravity so that's 32 .2 feet per square second for g times t squared.

01:39 And so from here we can see that we can find the time t for the ball to hit the ground is 0 .6822524 seconds.

01:54 We keep the decimal places for accuracy.

01:57 So if we go back to this initial equation, we can now find va since we know t...

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