# Time it takes for a rocket to reach maximum acceleration

1. Feb 22, 2012

### becky_marie11

1. The problem statement, all variables and given/known data
A rocket is launched vertically up with no initial velocity. Propulsion is provided by the ejection of mass with constant velocity of ejection u = 66.0 m/s relative to the rocket and at a constant rate so determined that the initial acceleration is zero. The mass of fuel that can be ejected is 94.0% of the total mass at launch. Assuming constant gravitational acceleration, how long does it take the rocket to achieve maximum upward acceleration?

u=66.0 m/s
Mass of fuel that can be ejected=94.0% of total mass at launch
g=9.81

2. Relevant equations
(1) ΔP=FgΔt
(2) Vf-Vi=uln(Mi/Mf
and maybe...
(3) Ru=Ma

3. The attempt at a solution
Alright, so first I found the delta t by plugging in the numbers given into (1). Then I solved for the final velocity using (2). Now I have no idea where to go and how to find the maximum acceleration and what time that happens at...just a point in the right direction would be nice!!!

2. Feb 23, 2012

### PeterO

I think we need to know the burn rate of the fuel - or at least the time it takes for all the fuel to burn, as then we may assume that half the fuel is gone in half that time.

My reasoning is as follows:

Suppose we were able to show that maximum acceleration is reached when half the fuel is gone [i don't believe that is true] then unless we know that the fuel takes a total of 90 seconds to burn, then we wouldn't know that max acceleration was at the 45 second mark.
If the fuel actually lasts 10 minutes, then max acceleration would be at 5 mins.

Perhaps it would be enough to say "when half the fuel is gone".

btw: I do expect that maximum acceleration is nearly 12g, provided the rocket hasn't got too far from the Earth by the time it happens.

Last edited: Feb 23, 2012
3. Feb 23, 2012

### Filip Larsen

The relative mass rate (that is, ejected mass per time divided by initial mass) can be found directly from (3). From this it is a simple matter to find out how long it takes to burn 94% of the mass.

4. Feb 23, 2012

### PeterO

You are clearly reading more into 3 than me, but then I am not aware what Ru is intended to stand for.

5. Feb 23, 2012

### Filip Larsen

I assume OP denotes mass flow rate as R, normally written as $\dot{m}$.

To expand a bit on my earlier post you can from the stated balance of thrust and weight at launch, $\dot{m}u=m_0 g$, and from the relationship between spend mass and time, $m_t = \dot{m}t$, easily find an expression for the spend mass ratio $m_t/m_0$ (the 94%) and solve for time.