What Exhaust Pipe Diameter is Needed for a Given Exhaust Speed?

[pizics]

Homework Statement

The engine turns at 3000 rpm at this speed. The engine capacity has a displacement of 1.2 litres: this means that 1.2 litres of exhaust gas is expelled from the motor (into the exhaust manifold) every two* revolutions. Further assume that the gas leaves the exhaust pipe with the same volume** that it leaves the engine. What exhaust pipe diameter is required to obtain the given exhaust speed?

Homework Equations

Remember the class problem about drag-race cars with exhaust thrust? Is there a serious propulsion effect for normal cars with horizontal exhaust pipes? To get quantitative, we need some chemistry. Here is an approximate reaction for burning hydrocarbons in air:

(CH2)n + 1.5*n O2 + 6n N2 -> n CO2 + n H2O + 6n N2 (n is an integer)

The (CH2)n approximates the fuel, and everything on the right is the exhaust. The atomic masses of hydrogen, carbon, nitrogen and oxygen are in the ratio 1:12:14:16. (Hint: how many kg of exhaust for one kg of fuel?)

A car uses 4.0 kg of (CH2)n hydrocarbon fuel per 100 km while it travels at 100 k.p.h. in a straight line at constant speed. The gas comes out of the (horizontal) exhaust pipe at a speed of v = 50 m.s-1, measured with respect to the car.

What is the magnitude of the force the exhaust gas exerts on the car? (Hint: the exhaust gasses are accelerated in the frame of the car, so ignore the momentum of the incoming gas.)

The Attempt at a Solution

I got V = 0.6R (R = revolutions), dV/dR = 0.6 And R = 50t, dR/dt= 50 dV/dt=30 I don't know what to do after this.

 

[haruspex]

There seem to be two parts to the question, the first being:

What exhaust pipe diameter is required to obtain the given exhaust speed?

But to answer it you need to use this piece of information provided in the second part:

The gas comes out of the (horizontal) exhaust pipe at a speed of v = 50 m.s-1

Where the second part is:

What is the magnitude of the force the exhaust gas exerts on the car? (

How can you relate the given value of v to your dV/dt and the diameter of the pipe?