A small rocket is fired in a test range. It rises high into the air and soon runs out of fuel. On the way down it passes near an observer (sitting in a 22.5-m-high tower) who sees the rocket traveling at a speed of 37.9 m/s and moving in a vertical plane at an angle of 50.8° below horizontal.
What is the maximum altitude of the rocket?
With what speed does the rocket hit the ground?
Once the rocket passes the observer, how long does it take to hit the ground?
vertical speed observed = -37.9 sin 50.8
= -29.4 m/s
how far did it fall to reach -29.4 m/s?
v = -g t
-29.4 = -9.81 t
t = 2.99 s
d = 4.9 t^2
= 4.9 (2.99)^2 = 43.9
so it reached 43.9 + 22.5 = 66.42 m
vertical speed at ground?
fell from 66.42 m
66.42 = 4.9 t^2
t = 3.68 s
v = -g t = -9.81 (3.68) = -36.1 m/s
additional time = 3.68-2.99
hits ground with initial velocity up reversed and same initial horizontal speed
u = 36.1 /tan 50.8 = 29.5
so speed at ground = sqrt u^2+v^2)
= sqrt (29.5^2+ 36.1^2)
= 46.6 m/s
To find the maximum altitude of the rocket, we need to analyze its motion.
First, we need to split the velocity of the rocket into horizontal and vertical components. The vertical component can be determined using trigonometry:
Vertical velocity = speed * sin(θ)
Vertical velocity = 37.9 m/s * sin(50.8°)
Vertical velocity = 29.89 m/s
The total time of flight can be calculated using the vertical motion of the rocket:
Time of flight = (2 * vertical velocity) / acceleration due to gravity
Time of flight = (2 * 29.89 m/s) / 9.8 m/s²
Time of flight ≈ 6.09 s
Now we can find the maximum altitude of the rocket:
Maximum altitude = initial vertical velocity * time of flight - (0.5 * acceleration due to gravity * (time of flight)²) + observer's height
Maximum altitude = 29.89 m/s * 6.09 s - (0.5 * 9.8 m/s² * (6.09 s)²) + 22.5 m
Maximum altitude ≈ 115.76 m
To find the speed with which the rocket hits the ground, we only need the horizontal component of the velocity since there is no horizontal acceleration:
Horizontal velocity = speed * cos(θ)
Horizontal velocity = 37.9 m/s * cos(50.8°)
Horizontal velocity = 24.23 m/s
Now, we can find the time it takes for the rocket to hit the ground after passing the observer. This can be determined using the vertical motion:
Time = (vertical velocity) / acceleration due to gravity
Time = 29.89 m/s / 9.8 m/s²
Time ≈ 3.05 s
Therefore, it takes approximately 3.05 seconds for the rocket to hit the ground after passing the observer.