# Work-kinetic energy theorem - model rocket velocity/height

• MonkeyDLuffy
In summary, the student calculates the work done by gravity on the rocket between 8.75 m and 75.0 m and uses this information to solve for the speed at 8.75 m.
MonkeyDLuffy

## Homework Statement

A student experimenting with model rockets measures the speed of a vertically-launched rocket to be 18.0 m/s when it is 75.0 m above the ground on the way up. The rocket engine fires from when the rocket is at ground level to when it is 8.75 m above the ground. If the rocket has a mass of 0.893 kg, use the work-kinetic energy theorem to determine:

(a) The speed of the rocket when it was 8.75 m above the ground
(b) The maximum height attained by the rocket

(make sure you know what the work-kinetic energy theorem is before you start your solution; it does not, for example, use gravitational potential energy.)

## Homework Equations

1. ΔK = ½m(V2 - V02)
2. Wnet = ΔK
3. Wgravity = mgh
4. V = V0 - gt2
5. ymax = y0 + V0t - ½gt2

## The Attempt at a Solution

a. My first instinct was to calculate ΔK for the entire flight, which I found to be about 145 J. Though I am not entirely sure this is true because the engine stopped firing at 8.75 m. What pushed me to believe this was the ΔK ,and therefore the Wnet, of the flight up to y = 75.0 m was a quote from my book which reads, " the work-energy theorem is valid even without the assumption that the net force is constant." Unfortunately, I cannot say with certainty that this is even relevant to the problem. If anyone could clarify this I would be very thankful.

b. That seemed to be a dead end, so I calculated the work done by gravity at 8.75 m and assumed this was the net-work from y = 0 m to y = 8.75 m. But after re-reading the problem I found that to be foolish, and so I was back at square one.

c. I became a bit distressed and decided to hit point a again, I figured this:

If Wnet = 145 J at y = 75.0 m , then the engine must have done upward work from y = 0 m to y = 8.75 m such that,
Wengine - Wgravity (which is about 657 J at y = 75.0 m) = 145 J.

So Wengine must be,

Wnet + Wgravity = 802 J.

If this were true, then:

Wengine = ΔK0→8.75m

Wengine = ½m(V2 - 0) → √(2Wengine)/m = V8.75m → V8.75m = 42.4 m/s

d. If this were all true, then the maximum height would be achieved when,

Wgravity = Wengine (a bit irrelevant but I'd like to know if this is true)

and I could use equation 4 above to solve for the time of the flight from y = 8.75 m to ymax, which I could then put into equation 5 where y0 = 8.75 m, and V0 = 42.4 m/s

I hope at least one of the conclusions I drew will be of actual use (though I'm doubtful). Thank you in advance for pointing me in the right direction!

Since you can solve an equation if it has only one unknown maybe you should apply the theorem between one point where the speed is known and one point where the speed is required.

Last edited:
So then I should calculate the work done by gravity on the rocket between 8.75 m and 75.0 m, and then set that equal ΔK to solve for the velocity at 8.75 m?

You are on the right track.

MonkeyDLuffy
That is how I thought one could solve for the speed of the rocket just as it stopped firing, yes.

MonkeyDLuffy
You are correct. The maximum velocity is achieved at the height that the rocket stopped firing.

MonkeyDLuffy
I've figured it out! Thanks for the input.

## 1. What is the work-kinetic energy theorem?

The work-kinetic energy theorem states that the net work done on an object is equal to the change in its kinetic energy. In other words, the work done on an object will result in a change in its velocity.

## 2. How does the work-kinetic energy theorem apply to model rockets?

In the context of model rockets, the work-kinetic energy theorem can be used to calculate the velocity and height of the rocket based on the force exerted on it and the distance it travels. This can help in designing and optimizing the performance of model rockets.

## 3. What factors affect the velocity and height of a model rocket?

The velocity and height of a model rocket are primarily affected by the force of the rocket's engine, the weight of the rocket, and the aerodynamics of its design. Other factors such as air resistance and wind conditions can also have an impact.

## 4. How can the work-kinetic energy theorem be used to improve the performance of model rockets?

By understanding the work-kinetic energy theorem, it is possible to make adjustments to the design and components of a model rocket to increase its velocity and height. For example, using a more powerful engine or reducing the weight of the rocket can result in higher velocities and heights.

## 5. Are there any limitations to using the work-kinetic energy theorem for model rockets?

While the work-kinetic energy theorem can provide a good estimate of the velocity and height of a model rocket, it does not take into account external factors such as air resistance and wind conditions. Therefore, the actual performance of a model rocket may differ from the calculations based on this theorem.

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