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Chapter 4: Problem 166

A rocket sled with initial mass of 3 metric tons, including 1 ton of fuel, rests on a level section of track. At \(t=\) \(0,\) the solid fuel of the rocket is ignited and the rocket burns fuel at the rate of \(75 \mathrm{kg} / \mathrm{s}\). The exit speed of the exhaust gas relative to the rocket is \(2500 \mathrm{m} / \mathrm{s}\), and the pressure is atmospheric. Neglecting friction and air resistance, calculate the acceleration and speed of the sled at \(t=10\) s.

Short Answer

Expert verified
The acceleration of the sled at \(t=10\) s is \(83.33 \, m/s^2\) and the speed is \(833.3 \, m/s\).

Step by step solution

01

Determine the force exerted by the rocket

According to the principle of conservation of momentum, the force exerted by the rocket can be calculated as the rate of change of momentum which is the mass of fuel burned per second multiplied by the exit velocity of the fuel. Given that \(75 \, kg\) of fuel are being exhausted each second at an exit speed of \(2500 \, m/s\), the force \(F\) exerted by the rocket can be found by multiplying these two values: \(F = 75 \, kg/s \cdot 2500 \, m/s = 187500 \, N\).
02

Calculate the mass of the rocket at given time

At \(t=10 \, s\), given that the sled burns fuel at a rate of \(75 \, kg/s\), the remaining mass would be the initial mass of the sled minus the weight of the burned fuel: \(m = 3000 \, kg - 10 \, s \cdot 75 \, kg/s = 2250 \, kg\).
03

Calculate the acceleration of the rocket

According to Newton's second law, the acceleration \(a\) of the object is given by the ratio of the force exerted on it and its mass, so \(a = F/m = 187500 \, N / 2250 \, kg = 83.33 \, m/s^2\).
04

Calculate the speed of the rocket

Now, given that the acceleration is constant over time, the speed \(v\) of the rocket at any specific time can be calculated as the product of acceleration and time, so \(v = a \cdot t = 83.33 \, m/s^2 \cdot 10 \, s = 833.3 \, m/s\).

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