Maximum Height of model rocket

In summary: So by finding the maximum time, we can then plug it back into h(t) to find the maximum height. In summary, to determine the maximum height reached by the rocket, we need to find the maximum time by setting h'(t) = 0 and solving for t. Then, we can plug this value back into h(t) to find the maximum height.
  • #1
DerekF
2
0

Homework Statement


A model rocket is launched with an initial velocity of 55 m/s. Its height as a function of time can be modeled by h(t) = -4.9t2 + 55t. Determine the maximum height reached by the rocket.

Homework Equations


h(t) = -4.9t2 + 55t

The Attempt at a Solution



h'(0) = 55, since h'(t) = -4.9(0) + 55 = 55 <-- Height reached within time interval 0 - 1 s

I am not sure how to approach this problem though I assume it has something to do with the initial velocity since it is included in the question. The quadratic formula was used in another problem but it isn't applicable here as far as I can tell. It sounds like I need to find the maximum time or t, in order to be able to find the maximum height. If this isn't possible, would the correct thing be to substitute arbitrary values for time until the height is no longer increasing or positive? If not, I have absolutely no idea.

Any help would be great. Thank you.* Edit *

Following the logic above it seems like the maximum height would be approximately 153.6 metres after 6 seconds. If this is correct, is there a proper way to express this or find a more precise value for t?
 
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  • #2
DerekF said:

Homework Statement


A model rocket is launched with an initial velocity of 55 m/s. Its height as a function of time can be modeled by h(t) = -4.9t2 + 55t. Determine the maximum height reached by the rocket.


Homework Equations


h(t) = -4.9t2 + 55t


The Attempt at a Solution



h'(0) = 55, since h'(t) = -4.9(0) + 55 = 55 <-- Height reached within time interval 0 - 1 s

I am not sure how to approach this problem though I assume it has something to do with the initial velocity since it is included in the question. The quadratic formula was used in another problem but it isn't applicable here as far as I can tell. It sounds like I need to find the maximum time or t, in order to be able to find the maximum height. If this isn't possible, would the correct thing be to substitute arbitrary values for time until the height is no longer increasing or positive? If not, I have absolutely no idea.

Any help would be great. Thank you.


* Edit *

Following the logic above it seems like the maximum height would be approximately 153.6 metres after 6 seconds. If this is correct, is there a proper way to express this or find a more precise value for t?

Step 1: plot a graph of the function h(t), to see what is happening.
Step 2: translate the insights from your graph into a formal statement.
 
  • #3
DerekF said:

Homework Equations


h(t) = -4.9t2 + 55t

The Attempt at a Solution



h'(0) = 55, since h'(t) = -4.9(0) + 55 = 55 <-- Height reached within time interval 0 - 1 s

The correct expression for the derivative of the height function is:

h'(t) = -9.8t + 55
 
  • #4
SteamKing said:
The correct expression for the derivative of the height function is:

h'(t) = -9.8t + 55

Yeah, oops. I had that written down on paper but messed up here.
 

FAQ: Maximum Height of model rocket

1. What factors affect the maximum height of a model rocket?

The maximum height of a model rocket is affected by several factors, including the weight of the rocket, the amount of thrust produced by the engine, the aerodynamics of the rocket, and the atmospheric conditions during flight.

2. How can I calculate the maximum height of my model rocket?

To calculate the maximum height of a model rocket, you can use the rocket equation, which takes into account the mass of the rocket, the mass of the engine, and the thrust produced by the engine. You can also use a flight simulation software or perform test launches to determine the maximum height.

3. Is there a limit to how high a model rocket can fly?

There is no specific limit to how high a model rocket can fly. However, the altitude of a model rocket is limited by the power of the engine, the aerodynamics of the rocket, and the atmospheric conditions. Additionally, some countries have regulations on the maximum altitude that model rockets can reach.

4. Can I increase the maximum height of my model rocket?

Yes, there are several ways to increase the maximum height of a model rocket. You can use a more powerful engine, reduce the weight of the rocket, improve the aerodynamics, and launch the rocket in favorable atmospheric conditions. Additionally, using a launch pad or a launch rod can also help increase the height of the rocket.

5. How accurate are the estimates for the maximum height of a model rocket?

The accuracy of the estimated maximum height of a model rocket depends on the accuracy of the inputs used in the calculations. The rocket equation and flight simulation software can provide fairly accurate estimates if the inputs are accurate. However, due to external factors such as wind and air density, there may be some variations between the estimated and actual maximum height of the rocket.

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