Rocket Equations
Conservation of MomentumMomentum is mass (m) times velocity (V). The conservation of momentum law states that momentum is conserved and can be stated mathematically as the sum of all momentum is zero:
I discussed an experiment where a bowling ball was thrown away from you, and you are sitting on a chair that rolls easily, then after throwing the bowling ball away you would start rolling away in the opposite direction. Mathematically we can apply the conservation of momentum equation to show this:
or rearranging the terms
Where m1 is your mass
V1 is your final velocity
m2 is the bowling ball's mass
V2 is the bowling ball's velocitySo for this example if you weight 180 lbs., the bowling ball weights 10 lbs., and you throw it a velocity of 10 feet per second then you would start rolling a little over 6 inches a second in the opposite direction:
The negative sign means that you would move in the opposite direction that you threw the bowling ball. Notice that you would roll backwards very slowly because you weight much more than a bowling ball and cannot throw the ball that fast.
Rockets work on the same principle only they throw the combusted propellant back very very fast. We can see this in another example. If a rocket weighs 100,000 lbs, and expends 500 lbs. of fuel in one second, and the exhaust gasses travel at 10,000 feet per second; then in that one second the rocket will increase velocity by 5 feet per second.
This isn't really a good example, because the rockets mass isn't constant (after all it's lost the 500 lbs of fuel hasn't it). Also a rocket will probably expend fuel a lot longer than one second. A better equation in addressed in "Change of Velocity" (below).
ThrustNewton's first law is that a body in motion will tend to stay in motion and a body at rest will tend to stay at rest. Put another way it takes force to cause a change in velocity.
Newtons second law addresses force. The more force on an object the more it accelerates, but the more massive it is the more it resists acceleration. Mathematically force (F) is the change (or differential) of momentum with respect to time, i.e. a change in momentum requires force.
Typically mass is constant and the change in velocity with respect to time is acceleration (a), so the equation can be expressed as force equals mass times acceleration, or:
However, for a rocket, if we look at the exhaust products, the mass isn't constant but changes with flow, however the exit velocity of that exhaust velocity is constant. Therefore for rockets we can say force equals the mass flow rate (m* ) times the exhaust velocity (Ve ), or;
This is the thrust of the rocket at any given time neglecting any effects from outside air pressure.Therefore, when we account for the effects of air pressure this equation becomes:
Where
Ae = The area at the exit plane of the nozzle
Pe = The rocket exhaust pressure at the nozzle exit plane
Pa = The outside atmosphere pressureFrom this equation we can see why the optimum expansion for a nozzle is when the pressure at the exit plane equals the atmospheric pressure . When the nozzle pressure is higher than the outside atmospheric pressure (underexpanded), the exhaust area is too small, and not all the energy has been converted to exhaust velocity so the thrust isn't optimized. Alternatively, when the nozzle pressure is lower than the outside atmospheric pressure (overexpanded) then the second term in the equation (i.e. Ae ( Pe - Pa ) ) becomes negative and actually causes a drag instead of a thrusting force.
Change in VelocityOn rockets, typically the propellant flow rate ( m* ) is constant and the exit velocity of the propellant (Ve ) is also constant, so the thrust is approximately constant (it may change some as the atmospheric pressure changes as the rocket climbs in altitude - however this term is less dominant than the F = m* Ve term). Even though the thrust is constant, the acceleration rate is typically increasing. Why? The propellant being burned and expelled from the rocket is decreasing the weight of the overall vehicle. At any instant in time F = m a, so rockets acceleration is equal to the thrust divided by the vehicle mass at that instant.
What is often critical is determining the total change in velocity of the rocket due to a specific thrust in a linear direction. Every change in orbit, or boost is a change in velocity. To even achieve orbit, a certain velocity must be reached.
Change in velocity, is often called or delta V ( d V).
The determine delta V we go back to Newton's second law
To get the two equations we have already solved
However this time mass is expressed as a function of time. Setting the two equations equal we get
Using calculus we can integrate this formula from the initial mass ( mo ) to the final mass (mf ) and get the change in velocity, delta V ( d V)
Or in English, the change in velocity is equal to the exit velocity of the rocket's exhaust times the natural logarithm of the initial rocket mass before the rocket started firing divided by the final rocket mass after the rocket was completed firing.
Specific ImpulseSpecific Impulse is a parameter used in evaluating engine performance. Specific impulse ( Isp ) is the rockets thrust, for force (F), divided by the mass flow rate ( m* ) times gravity (g), or
This parameter has the units of seconds and can be looked at two ways.
�������- Specific Impulse is a measure of how many pounds of thrust one pound or propellant can deliver in one second.
�������- Specific Impulse is a measure of how many seconds one pound of propellant can deliver one pound of thrust.Specific impulse is often used to characterize rocket engines. The higher the specific impulse the more thrust can be obtained for the same amount of fuel, or the less fuel can be used for the same amount of thrust.
Propellants are also characterized by specific impulse. In the case of propellants, specific impulse refers to the maximum theoretical specific impulse that could be delivered by a specific propellant combination (assuming a perfect engine). If you know an engine's specific impulse and mass flow rate you can calculate thrust by rearranging the definition;
Also, since thrust is equal to the mass flow rate times the exhaust velocity of the rocket.
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