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
. 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.
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.
To find the time of flight from launch to impact, we need to consider both the powered flight time and the time for the rocket to come back down. Here's how you can calculate it step-by-step:
Step 1: Calculate the time for powered flight.
Using the formula for motion in a straight line with constant acceleration:
Given:
Initial velocity (u) = 0 (since the rocket starts from rest)
Angle of launch (θ) = 55°
Acceleration (a) = 25.0 m/s²
Time (t) = 15.0 s
To find the final velocity (v) of the rocket during the powered flight, we can use the formula:
v = u + at
v = 0 + (25.0 m/s² × 15.0 s)
v = 375 m/s
Step 2: Calculate the total time of flight.
After the engines shut off, the motion of the rocket follows a parabolic path, which is governed by projectile motion. In this case, the time it takes for the rocket to reach the ground is equal to twice the time it took to reach the maximum height.
Since the rocket reaches its maximum height before coming back down, we can use the formula for projectile motion to find the total time of flight:
t_total = 2 × t_max_height
The formula for time to reach maximum height in projectile motion is:
t_max_height = (v_y / g)
Where:
v_y = vertical component of the final velocity
g = acceleration due to gravity (approximately 9.8 m/s²)
The vertical component of the final velocity (v_y) can be calculated using trigonometry:
v_y = v × sin(θ)
v_y = 375 m/s × sin(55°)
v_y = 375 m/s × 0.8192
v_y ≈ 307.2 m/s
Now we can calculate the total time of flight:
t_total = 2 × (v_y / g)
t_total = 2 × (307.2 m/s / 9.8 m/s²)
t_total = 2 × 31.33 s
t_total ≈ 62.66 s
Therefore, the time of flight from launch to impact is approximately 62.66 seconds.
To find the total time of flight from launch to impact, we need to consider two phases: the engine-on time and the engine-off time.
1. Engine-On Time:
During the engine-on time, the rocket moves in a straight line at an angle of 55.0 degrees above the horizontal. The acceleration during this time is 25.0 m/s^2.
To find the time taken during the engine-on time, we can use the equation of motion:
s = ut + 0.5at^2
Here, s is the distance traveled, u is the initial velocity (which is 0 m/s since the rocket starts from rest), a is the acceleration, and t is the time.
As we want to find the total time of flight, we need to find the time of flight during the engine-on time. We can find it by solving the equation for time.
s = ut + 0.5at^2
Using s = 0 (since we are calculating the time of flight),
0 = 0 + 0.5(25.0)t^2
0.5(25.0)t^2 = 0
t^2 = 0
Since the equation gives t^2 = 0, it means t = 0. So the time during the engine-on phase is 0 seconds.
2. Engine-Off Time:
After the engines shut off, the rocket follows a parabolic path back to earth, experiencing only the gravitational force. We can use simple projectile motion to find the time of flight during this phase.
The vertical motion of the rocket can be treated as free fall with an initial vertical velocity of 0 m/s and an acceleration due to gravity of -9.8 m/s^2.
The formula to find the time of flight for vertical motion is:
t = (2 * initial vertical velocity)/acceleration due to gravity
Using the given values:
t = (2 * 0)/(-9.8)
t = 0
As we can see, the time during the engine-off phase is also 0 seconds.
Total Time of Flight:
Since both the engine-on time and engine-off time are 0 seconds, the total time of flight from launch to impact is 0 seconds.
Therefore, the rocket impacts immediately after the engines shut off.