A catapult launches a rocket at an angle of 56.9° above the horizontal with an initial speed of 100 m/s. The rocket engine immediately starts a burn, and for 3.25 s the rocket moves along its initial line of motion with an acceleration of 28.1 m/s2. Then its engine fails, and the rocket proceeds to move in free-fall.
(a) Find the maximum altitude reached by the rocket.
(b) Find its total time of flight.
(c) Find its horizontal range.
PLEASE HELP ASAP!
Find the similar problem solution in previous post: "tina" , Wednesday, April 25, 2012 at 10:32pm
To find the maximum altitude reached by the rocket, we need to determine the time it takes for the rocket to reach its peak and then use that time to find the corresponding height.
Step 1: Find the time taken to reach the peak:
Using the equation kinematic equation:
vf = vi + at
We know:
vi (initial velocity) = 100 m/s (launch speed)
a (acceleration) = -9.8 m/s^2 (acceleration due to gravity)
vf (final velocity) = 0 m/s (at the peak)
Rearranging the equation, we get:
t = (vf - vi) / a
Substituting the given values, we get:
t = (0 - 100) / -9.8
t = 10.2 seconds
Step 2: Calculate the maximum altitude:
To find the maximum altitude, we can use the kinematic equation for displacement:
d = vit + (1/2)at^2
We know:
vi (initial velocity) = 100 m/s
a (acceleration) = -9.8 m/s^2
t (time) = 10.2 seconds
Substituting the given values, we get:
d = 100 * 10.2 + (1/2) * -9.8 * (10.2)^2
d = 510 - 500.4
d = 9.6 meters
Therefore, the maximum altitude reached by the rocket is 9.6 meters.
Now let's move on to finding the total time of flight.
Step 3: Calculate the total time of flight:
The total time of flight is the time from launch to landing. It will be the sum of the time taken to reach the peak (which we found in step 1) and the time taken to reach the ground again.
The time it takes to fall back to the ground is the time it takes for the rocket to reach its peak again, which is also 10.2 seconds.
Therefore, the total time of flight is:
Total time of flight = 2 * 10.2
Total time of flight = 20.4 seconds
So, the total time of flight is 20.4 seconds.
Next, let's find the horizontal range (the distance traveled by the rocket).
Step 4: Calculate the horizontal range:
The horizontal range is the distance traveled by the rocket in the horizontal direction. It can be found using the equation:
Range = horizontal velocity * time
The horizontal velocity can be calculated using:
velocity = initial velocity * cos(angle)
We know:
initial velocity = 100 m/s
angle = 56.9°
Substituting the given values, we get:
velocity = 100 * cos(56.9°)
velocity = 100 * 0.562
velocity = 56.2 m/s
Now we can calculate the horizontal range:
Range = 56.2 * 20.4
Range = 1144.8 meters
Therefore, the horizontal range of the rocket is 1144.8 meters.
To summarize:
(a) The maximum altitude reached by the rocket is 9.6 meters.
(b) The total time of flight is 20.4 seconds.
(c) The horizontal range is 1144.8 meters.