A rocket is fired at a speed of 66.0 m/s from ground level, at an angle of 63.0 ° above the horizontal. The rocket is fired toward an 31.2-m high wall, which is located 28.0 m away. The rocket attains its launch speed in a negligibly short period of time, after which its engines shut down and the rocket coasts. By how much does the rocket clear the top of the wall?
To find out how much the rocket clears the top of the wall, we need to calculate the maximum height it reaches.
Step 1: Calculate the rocket's initial vertical velocity.
The initial vertical velocity can be found using trigonometry. Since the rocket is launched at an angle of 63.0 ° above the horizontal, the vertical component of the initial velocity, v0y, is given by:
v0y = v0 * sin(θ)
where v0 is the launch speed and θ is the launch angle.
In this case, v0 = 66.0 m/s and θ = 63.0 °. Plug in these values to calculate v0y.
v0y = 66.0 m/s * sin(63.0 °) = 59.17 m/s
Step 2: Calculate the time taken to reach the maximum height.
The time taken to reach the maximum height can be found since we know the initial vertical velocity and the acceleration due to gravity. The acceleration due to gravity, g, is approximately 9.8 m/s².
The equation for motion in the vertical direction is:
y = v0y*t - (1/2)*g*t²
where y is the height, v0y is the initial vertical velocity, g is the acceleration due to gravity, and t is the time taken.
At the maximum height, the vertical velocity becomes zero, so we can set v0y*t - (1/2)*g*t² = 0 and solve for t.
(1/2)*g*t² = v0y*t
(1/2)*9.8 m/s² * t² = 59.17 m/s * t
(4.9 m/s²) * t = 59.17 m/s
t = 59.17 m/s / (4.9 m/s²) = 12.07 s (rounded to two decimal places)
Step 3: Calculate the maximum height reached by the rocket.
Using the time calculated in step 2, we can substitute it into the equation for height:
y = v0y*t - (1/2)*g*t²
y = 59.17 m/s * 12.07 s - (1/2)*9.8 m/s² * (12.07 s)²
y = 357.44 m - 714.88 m
y = -357.44 m (negative because it's measured downwards)
The negative value means that the rocket reaches a height of 357.44 m below the initial position (ground level), which is above the top of the wall.
Step 4: Find how much the rocket clears the top of the wall.
Since the wall is at a height of 31.2 m, we can subtract the height of the wall from the negative height calculated in step 3 to find the clearance.
Clearance = |y| - Height of the wall = |-357.44 m| - 31.2 m = 326.24 m
Therefore, the rocket clears the top of the wall by 326.24 meters.