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Chapter 4: Problem 55

What engine thrust (force) is needed to accelerate a rocket of mass \(m\) (a) downward at \(1.40 g\) near Earth's surface; (b) upward at \(1.40 g\) near Earth's surface; (c) at \(1.40 g\) in interstellar space, far from any star or planet?

Short Answer

Expert verified
In conclusion, the engine thrust for the different motions of the rocket are as follows (a) \(F = m(0.4g)\) (b) \(F = m(2.4g)\) and (c) \(F = m(1.4g)\). It should be noted that we've neglected air resistance in these calculations.

Step by step solution

01

Calculate Net Force for (a)

Let the engine thrust be \(F\) and the force due to gravity is \(mg\), where \(g = 9.81 ms^{-2}\) is the acceleration due to gravity. The rocket is accelerating downward at \(1.4g\), which means the net force \(F_{net}\) is the sum of the engine thrust and the weight of the rocket i.e., \(F_{net} = F + mg\). According to Newton's second law, \(F_{net} = ma\), where \(a = 1.4g\). Therefore, \(F + mg = m(1.4g)\). Solving this equation gives: \(F = m(1.4g) - mg\).
02

Calculate Net Force for (b)

Now, the rocket is accelerating upward at \(1.4g\). The engine thrust must overcome the force due to gravity before the rocket can accelerate upwards. So, the net force \(F_{net}\) is the difference between the engine thrust and the weight of the rocket i.e., \(F_{net} = F - mg\). Applying Newton's second law, we get \(F - mg = m(1.4g)\). Solving this equation gives: \(F = m(1.4g) + mg\).
03

Calculate Net Force for (c)

In interstellar space, far from any star or planet, there is no gravitational force acting on the rocket. Hence, the engine thrust is equal to the net force required to accelerate the rocket at \(1.4g\). Therefore, \(F = ma = m(1.4g)\).
04

Drawing conclusions

From the above steps, we can conclude that the engine thrust needed for each case depends on the direction of acceleration and the presence of gravitational forces. Near the Earth's surface, more engine thrust is required to accelerate the rocket upwards compared to downwards due to the force of gravity. In outer space, where gravity is negligible, the engine thrust is simply the product of the rocket's mass and the acceleration.

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