A sounding rocket, launched vertically upward with an initial speed of
79.0 m/s,
accelerates away from the launch pad at
5.70 m/s2.
The rocket exhausts its fuel, and its engine shuts down at an altitude of
1.15 km,
after which it moves freely under the influence of gravity.
a) How long is the rocket in the air?
s
b) What is the maximum altitude reached by the rocket?
km
(c) What is the velocity of the rocket just before it strikes the ground?
find the time it takes to get to 1.15km
1.15= +vi*t+5.7/2 t^2
use the quadratic equation to find t.
then find the time to free fly
hf=0=1.15+vb*t-1/2 g t^2
where vb= 79+5.7t under power.
now solve for t in the free fall.
add the two times.
To answer these questions, we can use the equations of motion. Let's break down the problem step by step.
a) How long is the rocket in the air?
To determine the time the rocket is in the air, we need to find the time it takes for the rocket to reach its maximum altitude. We can use the equation:
v = u + at
Where:
v = final velocity (0 m/s as the engine shuts down)
u = initial velocity (79.0 m/s)
a = acceleration (-5.70 m/s^2 since it's opposite to the rocket's initial velocity)
t = time taken
Rearranging the equation to solve for time (t):
t = (v - u) / a
Plugging in the values:
t = (0 - 79.0) / (-5.70) = 13.85 seconds (rounded to two decimal places)
Therefore, the rocket is in the air for approximately 13.85 seconds.
b) What is the maximum altitude reached by the rocket?
To find the maximum altitude reached by the rocket, we can use the equation of motion:
s = ut + (1/2)at^2
Where:
s = displacement (altitude)
u = initial velocity (79.0 m/s)
a = acceleration (-5.70 m/s^2)
t = time taken (13.85 seconds)
Plugging in the values:
s = (79.0)(13.85) + (1/2)(-5.70)(13.85)^2 = 535.3375 meters (rounded to two decimal places)
Therefore, the maximum altitude reached by the rocket is approximately 535.34 meters or 0.53534 kilometers.
c) What is the velocity of the rocket just before it strikes the ground?
Since the rocket moves freely under the influence of gravity after the engine shuts down, we can find its velocity just before it strikes the ground using the equation:
v^2 = u^2 + 2as
Where:
v = final velocity (what we need to find)
u = initial velocity (0 m/s as the engine shuts down)
a = acceleration due to gravity (-9.8 m/s^2, assuming downward direction)
s = displacement (negative since it's moving downward, equal to -1.15 km)
Rearranging the equation to solve for velocity (v):
v = √(u^2 + 2as)
Plugging in the values:
v = √((0)^2 + 2(-9.8)(-1.15)) ≈ √(22.54) ≈ 4.75 m/s
Therefore, the velocity of the rocket just before it strikes the ground is approximately 4.75 m/s.
To solve the problem, we can use the equations of motion for constant acceleration:
a) To find how long the rocket is in the air, we can use the equation:
t = (vf - vi) / a
where:
t = time
vf = final velocity
vi = initial velocity
a = acceleration
Initially, the rocket is moving upwards with an initial velocity of 79.0 m/s. The final velocity is 0 m/s because the rocket has stopped moving at the maximum altitude. The acceleration is 5.70 m/s².
Plugging in the values:
t = (0 - 79.0) / (-5.70)
t = 13.86 s
Therefore, the rocket is in the air for approximately 13.86 seconds.
b) To find the maximum altitude reached by the rocket, we can use the equation:
h = vi * t + (1/2) * a * t²
where:
h = maximum altitude
Using the time found in part (a), we substitute the values:
h = 79.0 * 13.86 + (1/2) * (-5.70) * (13.86)²
h ≈ 586.4 m
Therefore, the maximum altitude reached by the rocket is approximately 586.4 meters.
c) Since the rocket is freely moving under the influence of gravity after the engine shuts down, just before it strikes the ground, its velocity can be found using the equation:
vf = vi + a * t
where:
vf = final velocity
vi = initial velocity
a = acceleration
t = time
At the instant just before it strikes the ground, the final velocity will be the velocity of the rocket.
Plugging in the values:
vf = 79.0 + (-5.70) * 13.86
vf ≈ -6.99 m/s (approximated to two decimal places)
Therefore, the velocity of the rocket just before it strikes the ground is approximately -6.99 m/s.