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Chapter 8: Problem 42

A cart running on frictionless air tracks is propelled by a stream of water expelled by a gas-powered pressure washer stationed on the cart. There is a \(1.00-\mathrm{m}^{3}\) water tank on the cart to provide the water for the pressure washer. The mass of the cart, including the operator riding it, the pressure washer with its fuel, and the empty water tank, is \(400 . \mathrm{kg} .\) The water can be directed, by switching a valve, either backward or forward. In both directions, the pressure washer ejects \(200 .\) L of water per min with a muzzle velocity of \(25.0 \mathrm{~m} / \mathrm{s}\). a) If the cart starts from rest, after what time should the valve be switched from backward (forward thrust) to forward (backward thrust) for the cart to end up at rest? b) What is the mass of the cart at that time, and what is its velocity? (Hint: It is safe to neglect the decrease in mass due to the gas consumption of the gas-powered pressure washer!) c) What is the thrust of this "rocket"? d) What is the acceleration of the cart immediately before the valve is switched?

Short Answer

Expert verified
Determine the mass and velocity of the cart at that time, the thrust of the "rocket" (pressure washer), and its acceleration. Answer: To find the time at which the valve should be switched, the mass and velocity of the cart, the thrust of the "rocket", and its acceleration, follow the steps outlined in the solution. Use the conservation of momentum and Newton's second law to analyze the motion of the cart propelled by a pressure washer. Calculate the necessary variables using the mass flow rate, muzzle velocity, and remaining mass of the cart.

Step by step solution

01

Calculate the total momentum change for the cart to come to rest

To stop the cart, the momentum change caused by the water ejection must be equal to the initial momentum of the cart, which is zero as the cart starts from rest. Using the mass flow rate and the muzzle velocity, we can find the total momentum change: $$\text{Momentum Change} = \text{(Mass flow rate)} \times \text{(Muzzle velocity)}$$
02

Calculate the time needed for the momentum change to be achieved

We know that the mass flow rate is \(200\) L/min and the muzzle velocity is \(25.0\) m/s. To find the time needed for the momentum change to occur, we can divide the total momentum change by the mass flow rate and the muzzle velocity: $$\text{Time} = \frac{\text{Momentum Change}}{\text{(Mass flow rate)}\times\text{(Muzzle velocity)}}$$ #b. Mass and velocity of the cart#
03

Calculate the remaining mass of the cart

At the time the valve is switched, some amount of water has been ejected. To find the remaining mass of the cart, subtract the ejected water mass from the initial mass: $$\text{Remaining mass} = \text{Initial mass} - \text{(Ejected water mass)} $$
04

Calculate the velocity of the cart

As the ejected water provides a forward thrust to the cart, we can equate the momentum of the cart to the momentum change provided by the ejected water. The velocity of the cart can then be found by dividing the total momentum change by the remaining mass of the cart: $$\text{Velocity} = \frac{\text{Momentum Change}}{\text{Remaining mass}}$$ #c. Thrust of the "rocket"#
05

Calculate the force exerted by the water jet

The force exerted by the water jet can be found using the mass flow rate and the muzzle velocity: $$\text{Force} = \text{(Mass flow rate)} \times \text{(muzzle velocity)}$$ #d. Acceleration of the cart#
06

Calculate the acceleration of the cart

Using Newton's second law, the acceleration of the cart can be calculated by dividing the force exerted by the water jet by the remaining mass of the cart: $$\text{Acceleration} = \frac{\text{Force}}{\text{Remaining mass}}$$ By following these steps and using the given parameters, we can determine the time to switch the valve, the mass and velocity of the cart at that time, the thrust of the "rocket" (pressure washer), and its acceleration.

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