A rocket is launched from rest and moves in a straight line at 42.5 ° above the horizontal with an acceleration of 62.5 m/s2. After 32.0 s of powered flight, the engines shut off and the rocket follows a parabolic path back to earth. Find the time of flight from launch to impact.
What is the maximum altitude reached?
What is the horizontal distance between the launch pad and the impact point?
Jada/Fabio ~
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Rocket
To find the time of flight from launch to impact, we need to first calculate the time it takes for the rocket to reach its highest point during powered flight, and then add that to the total time of flight during the parabolic descent.
During powered flight, the rocket has an initial velocity of 0 m/s and an acceleration of 62.5 m/s^2. The angle of the launch path is given as 42.5° above the horizontal. We can use the equations of motion to find the time it takes to reach the maximum altitude.
1. Calculate the time to reach maximum altitude during powered flight:
Using the equation: v = u + at, where u is the initial velocity, a is the acceleration, t is the time, and v is the final velocity.
Since the final velocity at maximum altitude is 0 m/s, we can rearrange the equation to solve for time: t = (v - u) / a.
Substituting the values, we have:
t = (0 - 0) / 62.5
t = 0 s
This means the rocket reaches its maximum altitude instantly after the engines shut off.
2. Calculate the total time of flight during parabolic descent:
Given that the total time of flight from launch to impact is 32.0 s, and during powered flight the rocket reached its maximum altitude instantly, the remaining time must be during parabolic descent.
So, the total time of flight during parabolic descent is 32.0 s - 0 s = 32.0 s.
Therefore, the time of flight from launch to impact is 32.0 s.
To find the maximum altitude reached by the rocket, we can use the equations of motion for vertical motion. Considering the motion during parabolic descent, we know that at the maximum altitude, the final vertical velocity is 0 m/s.
3. Calculate the maximum altitude reached:
Using the equation: v^2 = u^2 + 2as, where u is the initial velocity, a is the acceleration, s is the displacement, and v is the final velocity.
Since the final velocity at maximum altitude is 0 m/s, we can rearrange the equation to solve for displacement: s = (v^2 - u^2) / (2a).
Substituting the values, we have:
s = (0^2 - 0^2) / (2 * 9.8)
s = 0 m
Therefore, the maximum altitude reached by the rocket is 0 m.
To find the horizontal distance between the launch pad and the impact point, we can use the equation for horizontal motion during parabolic descent.
4. Calculate the horizontal distance:
The horizontal distance during parabolic descent is the same as the horizontal distance during powered flight, as there are no horizontal forces acting on the rocket once the engines shut off.
Using the equation: s = ut + (1/2)at^2, where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time.
Substituting the values, we have:
s = 0 m/s * 32.0 s + (1/2) * 0 m/s^2 * (32.0 s)^2
s = 0 m
Therefore, the horizontal distance between the launch pad and the impact point is 0 m.