Determine the ratio of the times for the one-bounce and no-bounce throws

[Sunnie]

When baseball players throw the ball in from the outfield, they usually allow it to take one bounce before it reaches the infield, on the theory the ball arrives sooner that way. Suppose the angle at which a bounced ball leaves the ground is the same as the angle at which the outfielder threw it, as in the figure, 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 should the fielder throw the ball to make it go the same distance D with one bounce (blue path) as a ball thrown upward at 40.0° with no bounce (green path)?
°

(b) Determine the ratio of the times for the one-bounce and no-bounce throws.
(one-bounce time / no-bounce time)

Does anyone know how to do this problem. I have no formula's for this does anyone now the formula's?

 

[kamerling]

do you know how to compute the flight time and distance without bounces with a given initial speed an angle?

 

[Moetasim]

I can provide a response to this content. To determine the ratio of the times for the one-bounce and no-bounce throws, we first need to understand the concept of projectile motion. When an object, such as a baseball, is thrown at an angle, it follows a curved path known as a parabola. The time it takes for the object to reach the ground is determined by its initial velocity, angle of projection, and the acceleration due to gravity.

In this scenario, we are given that the angle of projection for both the one-bounce and no-bounce throws is the same. However, the speed of the ball after the bounce is one half of its initial speed. This means that the time it takes for the ball to reach the ground after the bounce will also be half of the time it takes for the ball to reach the ground without bouncing.

To answer part (a) of the question, we can use the formula for the range of a projectile, which is given by R = (v^2 * sin2θ)/g, where R is the range, v is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity. Since we are looking for the same distance D for both the one-bounce and no-bounce throws, we can set the ranges equal to each other and solve for the angle θ.

R(one-bounce) = R(no-bounce)
(v^2 * sin2θ)/g = (v^2 * sin40)/g
sin2θ = sin40
2θ = 40
θ = 20 degrees

Therefore, the angle at which the fielder should throw the ball is 20 degrees.

To answer part (b) of the question, we can use the formula for the time of flight of a projectile, which is given by t = (2v * sinθ)/g, where t is the time of flight. Since we know the initial velocity and angle of projection for both throws, we can calculate the time for each throw.

t(one-bounce) = (2 * v/2 * sin20)/g = (v * sin20)/g
t(no-bounce) = (2 * v * sin40)/g

The ratio of the times for the one-bounce and no-bounce throws is given by:
t(one-bounce)/t(no-bounce) = [(v * sin