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 = 32.0 m above sea level, directed at an angle theta = 47.3° above the horizontal, and with a speed v = 35.5 m/s. Assuming that air friction can be neglected, calculate the horizontal distance D traveled by the projectile.

See previous post: Tue, 12-1-15, 12:30 PM.

To calculate the horizontal distance traveled by the projectile, we can use the equations of motion.

First, let's break down the initial velocity of the rock into its horizontal and vertical components:

Initial vertical velocity (Vy) = v * sin(theta)
Vy = 35.5 m/s * sin(47.3°)
Vy = 35.5 m/s * 0.720
Vy = 25.56 m/s

Initial horizontal velocity (Vx) = v * cos(theta)
Vx = 35.5 m/s * cos(47.3°)
Vx = 35.5 m/s * 0.694
Vx = 24.61 m/s

Now, we can calculate the time of flight (t) using the vertical motion equation:
H = (1/2) * g * t^2
where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Rearranging the equation:
t^2 = (2H) / g
t^2 = (2 * 32.0 m) / 9.8 m/s^2
t^2 = 6.53 s^2
t = √(6.53 s^2)
t ≈ 2.56 s

Finally, we can calculate the horizontal distance using the equation:
D = Vx * t
D = 24.61 m/s * 2.56 s
D ≈ 63.02 m

Therefore, the horizontal distance traveled by the projectile is approximately 63.02 meters.

To find the horizontal distance traveled by the projectile, you can use the equations of motion for projectile motion.

First, we need to break the initial velocity of the projectile into horizontal and vertical components.

The horizontal component, Vx, is given by:

Vx = v * cos(theta)

Substituting the given values:

Vx = 35.5 m/s * cos(47.3°)

Next, we can find the time of flight, t, which is the time it takes for the projectile to reach the ground. We can find this using the vertical component of velocity and the acceleration due to gravity.

The vertical component of velocity, Vy, is given by:

Vy = v * sin(theta)

Substituting the given values:
Vy = 35.5 m/s * sin(47.3°)

The time of flight, t, can be found using the equation:

0 = Vy - g * t

Where g is the acceleration due to gravity, approximately 9.8 m/s².

Solving for t:
t = Vy / g

Next, we can find the horizontal distance traveled, D, which can be calculated using the equation:

D = Vx * t

Substituting the values we derived for Vx and t:

D = (35.5 m/s * cos(47.3°)) * (35.5 m/s * sin(47.3°)) / 9.8 m/s²

Calculating this expression will give us the horizontal distance traveled by the projectile.