A 1-kg rocket is fired straight up from level ground. The engine is able to provide a constant 15N of thrust for exactly 5 seconds after which it shuts of completely. Find the maximum height that the rocket achieves.
WEll, the impulse given is 15*5 Ns=75Ns
Velocity at burnout= 75Ns/1= 75m/s
What was its altditude at this point?
Vf^2=Vi^2+2ad
where a= (F/m -g)
solve for d (height hi)
Now, final height
Vf^2=0=75^2-2g(hf-hi)
solve for hf.
thanks so much...i got 33.3m/s...would u say that's correct?
sorry that was 188 that i got
To find the maximum height that the rocket achieves, we need to consider the forces acting on the rocket and apply Newton's laws of motion.
Step 1: Determine the net force acting on the rocket at any given time.
At the start of the rocket's motion, the only force acting on it is the weight due to gravity (mg), which is equal to the rocket's mass (1 kg) multiplied by the acceleration due to gravity (9.8 m/s^2). Therefore, the net force is -9.8 N (downward).
As the rocket moves upward, the thrust force provided by the engine acts in the opposite direction to gravity, reducing the net force acting on the rocket. The magnitude of the net force decreases as the rocket accelerates upward.
After 5 seconds, the engine shuts off completely, and the only force acting on the rocket is gravity (downward).
Step 2: Calculate the acceleration of the rocket during different time intervals.
During the first 5 seconds, while the engine is providing a constant 15 N of thrust, we can use Newton's second law (F = ma) to determine the acceleration. Rearranging the equation, we have a = F/m = 15 N / 1 kg = 15 m/s^2 (upward).
After the engine shuts off, the only force acting on the rocket is gravity, resulting in a constant downward acceleration of 9.8 m/s^2.
Step 3: Calculate the time taken to reach maximum height.
To find the time taken to reach maximum height, we need to determine when the rocket will stop moving upward and start falling down. This occurs when the upward acceleration due to the engine equals the downward acceleration due to gravity.
Using the equation a = Δv/Δt (where Δv is the change in velocity and Δt is the change in time), we can equate the acceleration due to the engine to the acceleration due to gravity:
15 m/s^2 = 9.8 m/s^2
Δv / Δt = 9.8 m/s^2
The acceleration due to gravity remains constant throughout the entire motion, so we can write the equation as:
15 m/s^2 = -9.8 m/s^2 (negative sign indicating downward direction)
Δv / Δt = -9.8 m/s^2
Simplifying this equation, we get:
Δt = Δv / -9.8 m/s^2
Δt = -15 m/s^2 / -9.8 m/s^2 (canceling out the unit of m/s^2)
Δt = 1.53 s
Therefore, it takes approximately 1.53 seconds for the rocket to reach maximum height.
Step 4: Calculate the maximum height reached by the rocket.
To find the maximum height, we need to calculate the displacement during the upward motion when the engine is active, and then add it to the displacement during the downward motion after the engine shuts off.
During the upward motion, the displacement (Δy) can be calculated using the following kinematic equation:
Δy = v0 * Δt + (1/2) * a * (Δt)^2
where v0 is the initial velocity, which is 0 m/s (since the rocket starts from rest):
Δy = 0 * 1.53 + (1/2) * 15 * (1.53)^2
Simplifying the equation, we get:
Δy = (1/2) * 15 * 2.3441
Δy = 17.4655 m
Therefore, during the upward motion, the rocket reaches a maximum height of approximately 17.47 meters.
Note: It is important to clarify that this solution assumes no air resistance and neglects any effects like air drag, changes in gravity with altitude, and the size of the rocket itself, which will affect the accuracy of the answer.