When baseball players throw the ball in from the outfield,...

When baseball players throw the ball in from the outfield, they usually allow it to take one bounce before it reaches the infielder on the theory that the ball arrives sooner that way. Suppose that the angle at which a bounced ball leaves the ground is the same as the angle at which the outfielder launched it, as in Figure $\mathrm{P} 4.54$, but that the ball's speed after the bounce is one half of what it was before the bounce. (a) Assuming the ball is always thrown with the same initial speed, at what angle $\theta$ should the ball be thrown in order to go the same distance $D$ with one bounce (blue path) as a ball thrown upward at $45.0^{\circ}$ with no bounce (green path)? (b) Determine the ratio of the times for the one-bounce and no-bounce throws.

Key Concepts

Projectile Motion

Projectile motion describes the two-dimensional motion of an object launched into the air, subject only to the acceleration due to gravity. In solving problems of projectile motion, the motion is broken down into horizontal and vertical components that are analyzed separately using kinematics. This framework is essential to understand trajectories, ranges, and flight times regardless of additional events like bounces.

Decomposition of Motion into Components

Breaking the initial velocity into horizontal and vertical components is fundamental in projectile motion. The horizontal component remains constant in the absence of air resistance, while the vertical component is influenced by gravitational acceleration. This decomposition allows us to derive relations for the distance traveled and time of flight from a given launch angle and speed.

Range Equation

The range of a projectile is the horizontal distance it covers during its flight. For motion without any interruptions, the range can be calculated using the launch speed, angle, and gravitational acceleration. This range equation is key in comparing different projectile trajectories, such as those with and without a bounce, or when adjusting the launch angle to achieve a specified distance.

Inelastic Bounce Dynamics

An inelastic bounce in projectile motion refers to a collision with the ground where the speed after impact is reduced by a known factor, even though the angle of reflection remains the same as the angle of incidence. Understanding this concept is crucial when analyzing trajectories that involve a bounce, as it affects both the subsequent range and the overall flight time of the projectile.

Time of Flight

The time of flight is the total duration that a projectile remains airborne. It depends on the vertical component of the initial velocity and the effects of gravity. In problems where the projectile undergoes a bounce with a reduction in speed, the time of flight for different segments must be calculated separately and then compared, which is important for determining ratios between different throwing techniques.

Transcript

0:00 Hello.

00:02 In baseball, there is a theory that if you shot the ball and let it bounce in halfway between the final destination, it will reach faster than shooting it straight forward as shown in the green path.

00:15 So here there is a comparison between the two methods.

00:18 And first, we are asked to calculate the angle at which the ball should be thrown given that when the ball bounces back, it bounces with half of the original speed.

00:30 And we're comparing this with the green path, which is shot at angle 45 degrees.

00:36 So let's start by writing the equation of the range here, which is v .i squared, sine 2 theta over g.

00:49 Let's start for getting the range of the green path.

00:53 For the green path, it's v .i squared divided by g, and sine 2 theta here is sine 90, which is always 1.

01:02 That's the range of the green path directly.

01:07 Now let's do it for the blue one.

01:10 The blue one consists of two portions, this first one, and this second.

01:15 So this is r1 plus r2.

01:21 R1 is vi squared sine 2 theta, which is the unknown we're searching for, divided by g.

01:30 And r2 is the same, but with the only difference that the velocity is, half of the original velocity.

01:39 So this is the vi in that case, divided by g.

01:44 We can take this out of the bracket, this two out of the bracket, so it became four in the denominator.

01:53 So we have vi squared, and here we have four.

01:55 Adding these together, we have the range of the blue path is 5 over 4 v .i squared sine 2 theta divided by g.

02:08 And this is our b.

02:13 But the range of the blue ball and the red and the green one is the same.

02:19 So these two quantities should be equal.

02:21 And from this equality, we can get the value of theta...

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