A rocket is fired at a speed of 54.0 m/s from ground level, at an angle of 63.0 ° above the horizontal. The rocket is fired toward an 52.8-m high wall, which is located 37.0 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?
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To find out how much the rocket clears the top of the wall, we need to split the problem into horizontal and vertical components. Let's start by analyzing the vertical motion.
1. Find the time of flight:
The time it takes for the rocket to reach the top of its trajectory can be determined using the vertical component of its initial velocity. The initial vertical velocity can be found by multiplying the launch speed (54.0 m/s) by the sine of the launch angle (63.0°).
Vertical component of the initial velocity = 54.0 m/s * sin(63.0°)
2. Determine the maximum height reached:
The maximum height reached by the rocket can be determined using the kinematic equation for vertical motion:
Final vertical velocity^2 = Initial vertical velocity^2 + 2 * acceleration * vertical displacement
Since the rocket reaches its maximum height, the final vertical velocity will be zero. Additionally, the vertical displacement will be the maximum height reached (52.8 m).
Solve the equation for the initial vertical velocity:
0 = (54.0 m/s * sin(63.0°))^2 + 2 * (-9.8 m/s^2) * 52.8 m
3. Calculate the time of flight:
Once you have the initial vertical velocity, you can use it to calculate the time it takes for the rocket to reach maximum height. Use the equation:
Final vertical velocity = Initial vertical velocity + acceleration * time
Since the final vertical velocity is zero at maximum height, solve the equation for time.
0 = (54.0 m/s * sin(63.0°)) + (-9.8 m/s^2) * time
Solve for time.
Now that we've found the time of flight, we can move on to the horizontal motion.
4. Find the horizontal velocity:
The horizontal component of the initial velocity can be found by multiplying the launch speed by the cosine of the launch angle.
Horizontal component of the initial velocity = 54.0 m/s * cos(63.0°)
5. Determine the horizontal displacement:
To find out how far the rocket travels horizontally, we need to use the time of flight calculated earlier and multiply it by the horizontal velocity.
Horizontal displacement = time of flight * horizontal velocity
6. Calculate the vertical displacement of the rocket above the wall:
Now that we have the horizontal displacement, we can calculate the vertical displacement above the wall. The vertical displacement is the height reached minus the height of the wall.
Vertical displacement above the wall = Max height - Height of the wall
Finally, we have calculated the vertical displacement above the wall, which represents how much the rocket clears the top of the wall.