A model rocket is launched straight upward with an initial speed of 30.0 m/s. It accelerates with a constant upward acceleration of 2.50 m/s2 until its engines stop at an altitude of 160 m.
A. what is the maximum height reached by the rocket?
B. how long does the rocket stay in the air?
First find the time and velocity when the rocket stops accelerating ("burnout"). Then write equations for altitude and velocity vs time for times after burnout.
Maximum altitude will be obtained at the time when the velocity is zero, after burnout.
Solve the altitude-time equation for the additional after-burnout time it takes for the rocket to hit the ground. Be sure to add the time spent accelerating to get the total time aloft.
Someone will gladly critique your work.
1103.4
A,h=23m B.56m
To find the maximum height reached by the rocket, we need to determine when the rocket reaches its peak and then calculate the altitude at that point.
Since the rocket accelerates upward, we can use the kinematic equation for motion in one dimension to find the time it takes to reach the peak. The equation is:
Vf = Vi + at
Where:
Vf = final velocity (when the rocket reaches its peak, the final velocity will be 0 m/s),
Vi = initial velocity (30.0 m/s),
a = acceleration (2.50 m/s^2),
t = time.
Rearranging the equation to solve for time:
t = (Vf - Vi) / a
Plugging in the values:
t = (0 - 30.0) / (-2.50)
t = 12.0 s
It takes the rocket 12.0 seconds to reach its peak. Now, we can use the kinematic equation for displacement to find the altitude at the peak. The equation is:
d = Vit + 0.5at^2
Where:
d = displacement (altitude),
Vi = initial velocity (30.0 m/s),
a = acceleration (2.50 m/s^2),
t = time (12.0 s).
Plugging in the values:
d = (30.0)(12.0) + 0.5(2.50)(12.0)^2
d = 180.0 + 0.5(2.50)(144.0)
d = 180.0 + 180.0
d = 360.0 m
Therefore, the maximum height reached by the rocket is 360.0 meters.
To determine how long the rocket stays in the air, we need to consider two parts of its flight: the time it takes to reach the peak and the time it takes to descend.
We already found that it takes 12.0 seconds for the rocket to reach its peak. Now, we need to find the time it takes for the rocket to descend from the peak to the ground.
Since the rocket is subject to free fall due to gravity, the acceleration of the rocket during descent is -9.8 m/s^2 (acceleration due to gravity).
Using the equation for time:
t = (Vf - Vi) / a
Where:
Vf = final velocity (when the rocket reaches the ground, the final velocity will be 0 m/s),
Vi = initial velocity (0 m/s, as the rocket starts from rest at the peak),
a = acceleration (-9.8 m/s^2),
t = time.
Rearranging the equation to solve for time:
t = (Vf - Vi) / a
t = (0 - 0) / (-9.8)
t = 0 s
The time for descent is 0 seconds because the final velocity is zero.
Therefore, the total time the rocket stays in the air is the time it takes to reach the peak (12.0 seconds) plus the time for descent (0 seconds), which is 12.0 seconds.