# Calculating Acceleration with Varying Mass: Tutorial & Exercises

• mraptor
In summary, the Tsiolkovsky rocket equation is a set of mathematical equations that describe the motion of rockets. It is based on the conservation of momentum, and it is used to calculate the motion of rockets in conditions where there is a force acting on them.
mraptor
hi,

I want to calculate how is acceleration changing if I have changing mass, but constant trust i.e. :

T = m*a

a = T / m

(I know it has to be calculus).
Then again I also wan't to be able to calculate displacement and velocities etc..
Trying to find somewhere on the internet a tutorial on equations of motion when the acceleration is varying.. but most of the time I find equations for constant-acceleration.
Do you have a good tutorial ? (don't point me to wikipedia, it is good as reference but not as tutorial)
I would like also to have some simple Exersises, so I can figure out how it is done in general.

thank you

You are on the right track. a = T/m and it takes calculus.

The derivitive of 1/m with respect to m is -1/m2

So for constant T, the derivitive of T/m with respect to m is -T/m2

The minus sign indicates that as m increases the quotient T/m decreases.

Nice.. ok now how can I calculate displacement or time taken to cross specific distans having this acceleration...
I suppose I can't use :

d = x + v*t + 1/2 a*t^2

because this is only valid for constant acceleration ?

You are correct. For a non-constant acceleration instead of computing the change in velocity by simply multiplying acceleration by time, you have to compute it by integrating acceleration over time using calculus.

Similarly, for a non-constant velocity you compute change in position by integrating velocity over time rather than simply multiplying velocity by time.

You end up with a double integral.

The first integration to compute velocity as a function of time results in:

http://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation

Conservation of momentum, mv, leads to (mv)'= m'v+ mv'= 0 (the ' indicates the derivative) if there is no external force. If there is a force, then we do not have conservartion of momentum but have (mv)'= m'v+ mv'= F.

## 1. How do you calculate acceleration with varying mass?

To calculate acceleration with varying mass, you need to use the formula a = F/m, where a is acceleration, F is the net force acting on the object, and m is the mass of the object. This formula can be applied to any situation where the mass of the object changes over time.

## 2. What is the relationship between mass and acceleration?

The relationship between mass and acceleration is inverse. This means that as the mass of an object increases, its acceleration decreases. This is because a greater force is required to accelerate a heavier object compared to a lighter object.

## 3. How do you incorporate changing mass into acceleration calculations?

To incorporate changing mass into acceleration calculations, you need to use the concept of "instantaneous mass". This means that you need to calculate the mass of the object at each moment in time and use that value in the acceleration formula. You can do this by dividing the total mass of the object by the time interval in which the mass changes.

## 4. Can you provide an example of calculating acceleration with varying mass?

Sure! Let's say we have a rocket that has a mass of 1000 kg at takeoff and 500 kg at a later point in time when fuel has been burned. If a net force of 2000 N is applied to the rocket, we can calculate its acceleration using the formula a = F/m. Plugging in the values, we get a = 2000 N / (1000 kg / t), where t is the time interval in which the mass changes. So if t = 10 seconds, the acceleration would be 2 m/s^2.

## 5. How can calculating acceleration with varying mass be applied in real-life situations?

Calculating acceleration with varying mass can be applied in many real-life situations, such as rocket launches, car racing, or even simple activities like throwing a ball. In all of these scenarios, the mass of the object is changing over time, and understanding the relationship between mass and acceleration is crucial in determining the outcome of the activity.

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