A rocket is fired at a speed of 92.0 m/s from ground level, at an angle of 33.0 ° above the horizontal. The rocket is fired toward an 13.4-m high wall, which is located 26.0 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?
So rocket never got propulsion after launch, becomes a projectile.
u=92.0 m/s, θ=33°
horizontal velocity (constant)
ux=u cos(θ)
Time to reach wall (in seconds)
t1=26.0/ux seconds
vertical initial velocity
uy=u sin(θ)
vertical height at time t
sy(t)=uy*t-(1/2)gt^2
Substitute t=t1 to find the vertical height, and subtract 13.4 m to find clearance.
To find out how much the rocket clears the top of the wall, we need to analyze its motion in both the horizontal and vertical directions.
Let's break down the rocket's motion into horizontal and vertical components:
Horizontal Component:
The initial horizontal velocity (Vx) can be calculated using the launch speed and the launch angle. We can use trigonometry to find the horizontal component of velocity:
Vx = V * cosθ
Vx = 92.0 m/s * cos(33.0°)
Vx = 76.92 m/s (rounded to two decimal places)
Vertical Component:
The initial vertical velocity (Vy) can also be calculated using the launch speed and the launch angle:
Vy = V * sinθ
Vy = 92.0 m/s * sin(33.0°)
Vy = 49.62 m/s (rounded to two decimal places)
Now, we can use kinematic equations to find the time it takes for the rocket to reach the wall (t) in the horizontal direction:
distance = velocity * time
26 m = 76.92 m/s * t
t = 26 m / 76.92 m/s
t ≈ 0.338 seconds (rounded to three decimal places)
Next, we can determine the rocket's vertical position (Y) at time (t) using the following equation:
Y = Voy * t + (1/2) * g * t^2
Y = (49.62 m/s) * (0.338 s) + (1/2) * (-9.8 m/s^2) * (0.338 s)^2
Y = 16.743 m (rounded to three decimal places)
Therefore, the rocket clears the top of the wall by approximately:
Clearance = Y - 13.4 m
Clearance = 16.743 m - 13.4 m
Clearance ≈ 3.34 m (rounded to two decimal places)
Therefore, the rocket clears the top of the wall by approximately 3.34 meters.