A catapult on a cliff launches a large round rock towards a ship on the ocean below. The rock leaves the catapult from a height H = 35.0 m above sea level, directed at an angle theta = 46.3° above the horizontal, and with a speed v = 35.8 m/s. Assuming that air friction can be neglected, calculate the horizontal distance D traveled by the projectile.
To calculate the horizontal distance D traveled by the projectile, we can break down the initial velocity into its horizontal and vertical components.
Given:
Height of the catapult from sea level, H = 35.0 m
Launch angle above the horizontal, theta = 46.3°
Launch speed, v = 35.8 m/s
The vertical component of velocity (Vy) can be calculated using the launch angle and launch speed:
Vy = v * sin(theta)
The time of flight (t) can be determined by dividing twice the vertical component of velocity by the acceleration due to gravity (g):
t = 2 * Vy / g
The horizontal component of velocity (Vx) can be found using the launch angle and launch speed:
Vx = v * cos(theta)
Using the time of flight, we can calculate the horizontal distance D traveled by the projectile:
D = Vx * t
Now, let's calculate the values step by step:
Given:
H = 35.0 m
theta = 46.3°
v = 35.8 m/s
g = 9.8 m/s^2 (acceleration due to gravity)
Step 1: Calculate Vy
Vy = v * sin(theta)
Vy = 35.8 m/s * sin(46.3°)
Step 2: Calculate t
t = 2 * Vy / g
t = 2 * (35.8 m/s * sin(46.3°)) / 9.8 m/s^2
Step 3: Calculate Vx
Vx = v * cos(theta)
Vx = 35.8 m/s * cos(46.3°)
Step 4: Calculate D
D = Vx * t
D = (35.8 m/s * cos(46.3°)) * (2 * (35.8 m/s * sin(46.3°)) / 9.8 m/s^2)
Simplifying the calculation will give us the result for D.