A rocket is fired at a speed of 75.0 m/s from ground level, at an angle of 67.6° above the horizontal. The rocket is fired toward an 11.0 m high wall, which is located 29.5 m away. By how much does the rocket clear the top of the wall?
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To find out how much the rocket clears the top of the wall, we first need to determine the vertical height the rocket reaches at a distance of 29.5 m away.
Let's break down the initial velocity of the rocket into horizontal and vertical components. The horizontal component remains constant throughout the motion, while the vertical component changes due to the influence of gravity.
The horizontal component of the velocity can be found using the equation:
Vx = V * cos(θ)
where Vx is the horizontal component of velocity, V is the initial velocity of the rocket (75.0 m/s), and θ is the angle of projection (67.6°).
Plugging in the values:
Vx = 75.0 * cos(67.6°)
Vx ≈ 31.5 m/s (rounded to one decimal place)
The time taken for the rocket to travel 29.5 m horizontally is given by:
time = distance / horizontal velocity
time = 29.5 m / 31.5 m/s
time ≈ 0.94 seconds (rounded to two decimal places)
Next, let's find the vertical component of the velocity (Vy) at this time by using the equation:
Vy = V * sin(θ) - g * t
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and t is the time taken (0.94 seconds).
Plugging in the values:
Vy = 75.0 * sin(67.6°) - 9.8 * 0.94
Vy ≈ 53.8 - 9.2
Vy ≈ 44.6 m/s (rounded to one decimal place)
Now, we can find the vertical distance (height) the rocket reaches at a horizontal distance of 29.5 m by using the equation:
height = initial vertical velocity * time - (1/2) * g * t^2
Plugging in the values:
height = 44.6 * 0.94 - (1/2) * 9.8 * (0.94)^2
height ≈ 41.9 - 4.9 * (0.94)^2
height ≈ 41.9 - 4.9 * 0.88
height ≈ 41.9 - 4.3
height ≈ 37.6 m (rounded to one decimal place)
Finally, to determine how much the rocket clears the top of the wall, we subtract the height of the wall (11.0 m) from the maximum height reached by the rocket:
clearance = maximum height - height of the wall
clearance = 37.6 m - 11.0 m
clearance ≈ 26.6 m (rounded to one decimal place)
Therefore, the rocket clears the top of the wall by approximately 26.6 meters.