A tennis player wants to slam a serve at O so that the ball lands just inside the opposite corner of the court. What should the ratio be for the initial velocity ? The time t = 0 is the time when the ball is hit by the racket.
A) W/L
B) L/W

Question

A tennis player wants to slam a serve at O so that the ball lands just inside the opposite corner of the court. What should the ratio  be for the initial velocity  ? The time t = 0 is the time when the ball is hit by the racket. 
A) W/L 
B) L/W

Quizplus · Accepted Answer

If the ball lands just inside the opposite corner of the court, it means that the ball travels the diagonal of the court, which is equal to the hypotenuse of a right triangle formed by the width and length of the court. Let's call the width W and the length L.
So, we have:
h^2 = W^2 + L^2
where h is the length of the hypotenuse, which is the distance that the ball travels.
We want the ball to land just inside the opposite corner, which means the ball has to travel the diagonal minus the diameter of the ball, which is 2d, where d is the diameter.
So, h - 2d = W + L
We know that the initial velocity of the ball can be decomposed into its horizontal and vertical components:
Vx = Vo*cos(theta)
Vy = Vo*sin(theta)
where Vo is the initial velocity of the ball, theta is the angle of the serve, Vx is the horizontal component of the velocity, and Vy is the vertical component of the velocity.
The time it takes for the ball to reach the other side of the court is the same time it takes for the ball to reach the apex of its trajectory, where Vy = 0. Let's call this time t. We can find t by using the equation:
Vy = Vyo - g*t
where g is the acceleration due to gravity and Vyo is the initial vertical velocity, which is equal to Vo*sin(theta).
When Vy = 0, we have:
t = Vyo/g = Vo*sin(theta)/g
Now, we can use the time t to find the distance the ball travels horizontally, which is equal to the product of the horizontal component of the velocity and the time:
dx = Vx*t = Vo*cos(theta)*t
Substituting t with Vo*sin(theta)/g, we get:
dx = (Vo^2/g)*sin(theta)*cos(theta)
Now, we can use dx to find the ratio W/L:
(W+dx)/L = tan(alpha)
where alpha is the angle between the hypotenuse and the length of the court:
tan(alpha) = W/L
Substituting dx with (Vo^2/g)*sin(theta)*cos(theta), we get:
(W + (Vo^2/g)*sin(theta)*cos(theta)) / L = tan(alpha)
Simplifying, we get:
(W/L) + ((Vo^2/g)*sin(theta)*cos(theta)) / (gL) = tan(alpha)
Since we want to find the ratio Vo/W, let's solve this equation for that ratio:
(Vo/W) = (gL / (sin(theta)*cos(theta))) * (tan(alpha) - (W/L))
Substituting tan(alpha) with h/L, we get:
(Vo/W) = (gL / (sin(theta)*cos(theta))) * ((h/L) - 1)
Substituting h^2 = W^2 + L^2, we get:
(Vo/W) = (gL / (sin(theta)*cos(theta))) * (((W^2 + L^2)/L^2) - 1)^0.5 - 1
The only answer choice that matches this expression is B.

A Tennis Player Wants to Slam a Serve at O