A batter hits a ball, causing it to leave the bat at 135 m/s and at an angle of 18.3 degrees from the horizontal. If the baseball field is level, and ignoring the height at which the batter hits the ball, how far (horizontally, in meters) away from the batter does the ball hit the ground? Use g = 9.8 m/s2; take up and forward as positive and express your result to three significant digits.

Vi = initial speed up

u = constant horizontal speed
find t, time in air

Vi = 135 sin 18.3 = 39.25
u = 135 cos 18.3 = 128.2

d = u T

find T the time in air

v = Vi - 9.8 t
at top where v = 0 and t=T/2
0 = 39.25 - 9.8 t
t = 4.005 s
so T = 2 t = 8.01 seconds in the air
d = u t = 128.2 * 8.01 = 1027
which is 1030 to 3 figures

To determine how far the ball travels horizontally before hitting the ground, we need to analyze the motion of the ball in the horizontal and vertical directions separately.

First, let's break down the initial velocity of the ball into horizontal and vertical components. We can use trigonometry to find these components:

Horizontal component of velocity (Vx):
Vx = V * cos(θ)
Vx = 135 * cos(18.3°)
Vx ≈ 128.759 m/s

Vertical component of velocity (Vy):
Vy = V * sin(θ)
Vy = 135 * sin(18.3°)
Vy ≈ 45.995 m/s

Now, we can determine the time it takes for the ball to hit the ground. Since we are ignoring the height at which the batter hits the ball, we only need to consider the vertical motion. We can use the equation of motion:

h = Vy * t - (1/2) * g * t^2

Where h is the initial vertical displacement, Vy is the initial vertical velocity, g is the acceleration due to gravity, and t is the time.

Since the ball hits the ground, we know that the final vertical position (h) is 0. Assuming the initial vertical position is also 0, we can set the equation equal to zero:

0 = Vy * t - (1/2) * g * t^2

Rearranging the equation, we get:

(1/2) * g * t^2 = Vy * t

Simplifying:

(1/2) * g * t = Vy

Substituting the values:

(1/2) * 9.8 * t ≈ 45.995

Solving for t:

t ≈ 9.4 seconds

Finally, we can calculate the horizontal distance traveled by the ball using the horizontal velocity and the time it takes to hit the ground:

Horizontal Distance = Vx * t
Horizontal Distance ≈ 128.759 * 9.4
Horizontal Distance ≈ 1210.201 meters

Therefore, the ball will hit the ground approximately 1210.201 meters away from the batter.