A ballplayer standing at homeplate hits a baseball that is caught by another player at the same height above the ground from which it was hit. The ball is hit with an initial velocity of 23.0 m/s at an angle of 54.0° above the horizontal.

How high will the ball rise?
. m higher than where it was hit

(b) How much time will elapse from the time the ball leaves the bat until it reaches the fielder?
s

(c) At what distance from home plate will the fielder be when he catches the ball?

Vo = 23m/s @ 54o.

Xo = 23*cos54 = 13.52 m/s = Hor. component of initial velocity.
Yo = 23*sin54 = 18.61 m/s = Ver. component of initial velocity.

a. Y^2 = Yo^2 + 2g*h.
h = (Y^2-Yo^2)/2g=(0-346.3)/-19.6=17.7 m

b. Y = Yo + g*t.
Tr = (Y-Yo)/g = (0-18.61)/-9.8=1.90 s.
= Rise time.
Tf = Tr = 1.90 s. = Fall time.

T = Tr + Tf = 1.9 +\1.9 = 3.8 s.

c. d = Xo * T = 13.52 * 3.8 = 51.4 m.

To solve this problem, we can break it down into three parts:

(a) Finding the height the ball will rise to.
(b) Calculating the time taken for the ball to reach the fielder.
(c) Determining the distance from home plate to where the fielder catches the ball.

(a) To find the height the ball will rise, we need to analyze the vertical component of its initial velocity. The initial vertical velocity (Vy) can be found using the equation:

Vy = V * sin(θ)

where V is the initial velocity of the ball (23.0 m/s) and θ is the angle of projection (54.0°).

Vy = 23.0 m/s * sin(54.0°)
Vy = 18.49 m/s

Now, we can use the equations of motion to find the height reached by the ball. Assuming the initial vertical displacement (y) is zero, the equation for vertical displacement can be written as:

y = (Vy^2) / (2 * g)

where g is the acceleration due to gravity (9.8 m/s^2).

y = (18.49 m/s)^2 / (2 * 9.8 m/s^2)
y = 17.24 m

Therefore, the ball will rise 17.24 m higher than where it was hit.

(b) To determine the time it takes for the ball to reach the fielder, we can use the equation for time of flight:

T = 2 * Vy / g

T = 2 * 18.49 m/s / 9.8 m/s^2
T = 3.77 s

The time taken for the ball to reach the fielder is 3.77 seconds.

(c) Lastly, to find the distance from home plate to where the fielder catches the ball, we can calculate the horizontal displacement. The horizontal velocity (Vx) can be found using the equation:

Vx = V * cos(θ)

Vx = 23.0 m/s * cos(54.0°)
Vx = 12.57 m/s

Now, we can use the equation for horizontal displacement:

x = Vx * T

x = 12.57 m/s * 3.77 s
x ≈ 47.37 m

Therefore, the fielder will catch the ball at a distance of approximately 47.37 meters from home plate.