Consider motion of an object in a straight line. The following equation applies:

Work done on object (W) = Force (F) applied in the direction of travel * displacement (x)

Consider the object of mass 1000 kg initially at rest. A constant force is applied. For simplicity, assume that frictionless motion.

Initial velocity, v0 = 0.

Acceleration, a, is constant (constant force) = 1 m/s2

F = 1000 N

Consider first period of 10 seconds.

t1 = 10 seconds

x1 = 50 m

Work done in first 10 seconds, W1 = 1000 * 50 = 50 000 J.

v1 = 10 m/s

Kinetic energy = 0.5 * 1000 * 10^2 = 50 000 J.

So, all work done has been converted to kinetic energy.

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Consider second interval of 10 seconds.

t2 = 20 seconds

t2 - t1 = 10 s

Distance travelled during 2nd interval, x2 = 150 m

Work done in 2nd interval, W2 = 1000 * 150 = 150 000 J.

v2 = 20 m/s

Kinetic energy = 0.5 * 1000 * 20^2 = 200 000 J.

This equals W1 + W2.

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All very simple so far!

Now, this constant force was the constant thrust of a rocket engine. Thus the W1 and W2 was done by the rocket.

Since thrust was constant, the same amount of fuel was consumed in the 1st 10 seconds as the 2nd interval of 10 seconds. (A)

If the rocket's efficiency of converting fuel energy into work is assumed to be constant at 0.5.

Fuel energy in 1st interval = W1 / 0.5 = 50 000 / 0.5 = 100 000 J

Fuel energy in 2nd interval = W2 / 0.5 = 150 000 / 0.5 = 300 000 J

This contradicts (A) and common sense, which states that the rate of fuel burnt is constant!!!

What is your explanation of this paradox?

Doug