A rocket is launched from rest and moves in a straight line at 55.0 degrees above the horizontal with an acceleration of 25.0

Question

. After 15.0
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. HINT: Simple projectile motion after engines are shut down.
HINT: Do not forget to include the engine-on time in your time calculation.

Answer 1

To find the time of flight from launch to impact, we need to consider two separate periods: the powered flight period and the projectile motion period.

Powered Flight Period:
Given:
Initial velocity (V₀) = 0 (the rocket is launched from rest)
Acceleration (a) = 25.0°

Using the equations of motion, we can calculate the time it takes for the rocket to complete the powered flight period:
V = V₀ + at,
where V is the final velocity after time t.

Since we know the angle of launch is 55.0°, we can decompose the acceleration into its horizontal and vertical components:
a_horizontal = a * cos(55.0°),
a_vertical = a * sin(55.0°).

During the powered flight, the horizontal acceleration remains constant since we assume no air resistance. Therefore, the horizontal velocity (V_horizontal) at the end of the powered flight can be calculated as:
V_horizontal = a_horizontal * t.

For the vertical component, we know the initial velocity in the vertical direction is 0, as the rocket is launched from rest vertically. Therefore, the vertical velocity (V_vertical) at the end of the powered flight can be calculated as:
V_vertical = a_vertical * t.

Now, we can calculate the total velocity (V_total) at the end of the powered flight using the Pythagorean theorem:
V_total = sqrt(V_horizontal² + V_vertical²).

Since the rocket's engines shut off after 15.0 seconds, the time for the powered flight period is 15.0 seconds.

Projectile Motion Period:
During the projectile motion period, the rocket follows a parabolic path back to earth. We can use the kinematic equation for vertical displacement to find the total time of flight during this period:
h = V₀ * t + (1/2) * g * t²,
where h is the vertical displacement (the net change in height), V₀ is the initial vertical velocity, t is the total time of flight for the projectile motion, and g is the acceleration due to gravity (-9.8 m/s²).

We know the initial vertical velocity is V_vertical calculated during the powered flight period, and the final vertical displacement is 0 since the rocket impacts the ground. Therefore, we can rearrange the equation to solve for time (t):
(1/2) * g * t² + V_vertical * t = 0.

Once we find the total time during the projectile motion period, we can add it to the time during the powered flight period to get the total time of flight from launch to impact.

Please note that we are assuming no air resistance and disregarding any other external factors, which may not be the case in reality.