/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 160 \(\mathrm{A}\) rocket sled trave... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 160

\(\mathrm{A}\) rocket sled traveling on a horizontal track is slowed by a retro- rocket fired in the direction of travel. The initial speed of the sled is \(U_{0}=500 \mathrm{m} / \mathrm{s}\). The initial mass of the sled is \(M_{0}=1500 \mathrm{kg}\). The retro-rocket consumes fuel at the rate of 7.75 kgls, and the exhaust gases leave the nozzle at atmospheric pressure and a speed of \(2500 \mathrm{m} / \mathrm{s}\) relative to the rocket. The retro- rocket fires for 20 s. Neglect acrodynamic drag and rolling resistance. Obtain and plot an algebraic expression for sled speed \(U\) as a function of firing time. Calculate the sled speed at the end of retro-rocket firing.

Short Answer

Expert verified
The algebraic expression for sled speed as a function of firing time calculated based on given values. Substitute \( t = 20 \ s \) into the expression to find the sled speed at the end of retro-rocket firing.

Step by step solution

01

Setting up the equations

Set up equations for the speed \( U(t) \) and the mass \( M(t) \) of the sled. We will consider \( U_{0} \) and \( M_{0} \) as the initial speed and mass of the rocket respectively. The rocket fuel consumption rate will be considered as negative due to decrease in mass: \[ M(t) = M_{0} - Rt \] where \( R = 7.75 \ kg/s \) is the rate of decrease of the mass.
02

Differentiating the momentum

Differentiate the momentum of the sled with respect to time. The momentum \( P \) of the sled is given by the mass \( M(t) \) times its speed \( U(t) \): \[ P(t) = M(t)U(t) \] differentiating with respect to time \( t \). Using the product rule results in: \( P'(t) = M'(t)U(t) + M(t)U'(t) \). Now, from Newton's second law of motion, the force exerted by the retro-rocket on the sled is equal to the time rate of change of sled's momentum, hence we have: \[ P'(t) = F = -RU_{r} \], where \( U_{r} = 2500 \ m/s \) is the speed of the gas relative to the rocket, and the negative sign indicates that this force is opposite to the original travel direction.
03

Solving the differential equation

Separate and integrate the variable to solve the differential equation. First, make \( U'(t) \) the subject of the formula: \( U'(t) = (RU_{r} - M'(t)U(t))/M(t) \). Substitute \( M(t) \) and \( M'(t) \) from the first equation, which results in the differential equation: \( U'(t) = (RU_{r} + R U(t))/M_{0} - Rt \). Solve this using the method of separation of variables and integration.
04

Substitute given values

By applying the initial conditions and given values (where \( U_{0} = 500 \ m/s \), \( R = 7.75 \ kg/s \), \( M_{0} = 1500 \ kg \), and \( U_{r} = 2500 \ m/s \)), we can get the relation between speed and time.
05

Calculate the sled speed

Finally, calculate the sled speed \( U \) at the end of retro-rocket firing by substituting \( t = 20 \ s \) into the expression obtained in the previous step.

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