# Rocket Problem: Finding Velocity of Rocket as Function of Time

• jgens
In summary, the question is asking for the velocity of a rocket ascending in a uniform gravitational field with exhaust ejection and air resistance. The rocket equation can be used to solve for acceleration as a function of time, which can then be integrated twice to obtain the velocity as a function of time. The quadratic formula must be used when integrating to account for the air resistance force being proportional to the square of the velocity.
jgens
Gold Member

## Homework Statement

A rocket ascends from rest in a uniform gravitational field by ejecting exhaust with constant speed u. Assume that the rate at which mass is expelled is given by dm/dt = km, where m is the instantaneous mass of the rocket and k is a constant, and that the rocket is retaded by air resistance with a force bv, where b is a constant. Find the velocity of the rocket as a function of time.

## Homework Equations

FNET = m*dv/dt - u*dm/dt

## The Attempt at a Solution

I've tried writing m as a function of t and using that in the rocket equation to solve for v, but I keep getting a really nasty differential equation. This course doesn't presuppose much knowledge about DEs, so I think that I must be doing something wrong. Can anyone point me in the right direction?

Thank you for your question. It seems like you are on the right track with using the rocket equation to solve for the velocity of the rocket. However, there are a few things that you may want to consider in your approach.

Firstly, it may be helpful to rewrite the rocket equation in terms of acceleration instead of velocity. This can be done by using the chain rule and substituting in the given expressions for dm/dt and FNET. This will result in a differential equation that can be solved for the acceleration as a function of time.

After obtaining the acceleration, you can then integrate it twice to get the velocity as a function of time. Make sure to include the initial conditions of the rocket (starting from rest) in your integration.

Also, keep in mind that the air resistance force is proportional to the square of the velocity, so you will need to use the quadratic formula when integrating to account for the negative sign in the equation.

I hope this helps guide you in the right direction. Good luck with your problem-solving!

## 1. How do you find the velocity of a rocket as a function of time?

To find the velocity of a rocket as a function of time, you would need to use the equation v(t) = u + at, where v is the velocity, u is the initial velocity, a is the acceleration, and t is the time. You would also need to have data on the rocket's initial velocity and acceleration at different points in time.

## 2. What factors affect the velocity of a rocket?

The velocity of a rocket can be affected by various factors, such as the amount of thrust produced by the rocket's engine, the weight of the rocket, air resistance, and gravity. Changes in these factors can impact the rocket's acceleration and ultimately its velocity.

## 3. Can the velocity of a rocket change over time?

Yes, the velocity of a rocket can change over time. This is because the rocket's acceleration can change due to factors such as changes in thrust, weight, and air resistance. As the acceleration changes, the velocity will also change accordingly.

## 4. How can I graph the velocity of a rocket as a function of time?

To graph the velocity of a rocket as a function of time, you would plot the time on the x-axis and the velocity on the y-axis. You can use the equation v(t) = u + at to calculate the velocity at different points in time and then plot those points on the graph. This will give you a curve that represents the velocity of the rocket over time.

## 5. Why is it important to know the velocity of a rocket as a function of time?

Knowing the velocity of a rocket as a function of time is important for understanding the motion of the rocket and predicting its trajectory. This information is also crucial for ensuring the rocket reaches its intended destination and for making any necessary adjustments during the flight. It can also provide valuable data for future rocket designs and missions.

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