Question

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

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

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: $$\text{Momentum Change} = \text{(Mass flow rate)} \times \text{(Muzzle velocity)}$$

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#

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: $$\text{Remaining mass} = \text{Initial mass} - \text{(Ejected water mass)} $$

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"#

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#

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.

Key concepts

Momentum Change

Understanding the concept of momentum change is crucial when analyzing systems where forces are applied over time, such as a cart propelled by a stream of water. Momentum, defined as the product of mass and velocity, helps us describe the motion of an object.
In the provided exercise, the cart starts from rest, implying that its initial momentum is zero. To bring the cart to rest again after it has gained momentum from the backward ejection of water, the total momentum change must equal the initial momentum, which is still zero. This means the forward thrust by the water needs to precisely counterbalance the backward momentum gained.

• Momentum Change is given by: \[\text{Momentum Change} = \text{(Mass flow rate)} \times \text{(Muzzle velocity)}\]
• The muzzle velocity here is the speed at which water is ejected
• The mass flow rate is the volume of water ejected per unit time

By understanding this, we acknowledge that the total impact of water ejection is a fundamental way to influence and control the cart's motion.

Thrust Calculation

Thrust in the context of a propulsion system, like our cart's pressure washer, is defined as the force exerted in response to the expulsion of mass. It is calculated by multiplying the mass flow rate of the ejected water by the muzzle velocity at which it is ejected.
This force pushes the cart in the opposite direction of the expelled water, illustrating Newton’s third law: for every action, there is an equal and opposite reaction.

• Thrust Force is calculated as: \[\text{Force} = \text{(Mass flow rate)} \times \text{(muzzle velocity)}\]
• Represents the 'rocket' action of the pressure washer on the cart
• Key for determining how fast and forcefully the cart moves from its stationary position

By calculating this force, we understand how powerful our propulsion mechanism is in terms of moving the cart forward or backward.

Newton's Second Law

Newton's second law relates to the concept of force, mass, and acceleration. It tells us that the acceleration of an object depends on the net force acting on it and its mass. In a mathematical format, Newton’s second law is articulated as: \[F = ma\]
In the case of our cart, the force is the thrust calculated from the expelled water, and it acts on the entire system which includes the cart's mass and the water still contained within.

• The law helps in determining how the equilibrium of forces results in the motion of an object.
• By applying it here, we can assess how effectively the cart can be accelerated using the force generated by water expulsion.
• This is crucial when deciding how long the cart should expel water to reach certain speeds.

By using Newton's second law, we draw a link between the exerted force through our pressure washer and the resultant acceleration of the cart.

Acceleration Calculation

The acceleration of the cart gives us insight into how quickly it can change its velocity when thrust is applied. Acceleration depends not only on the thrust but also significantly on the remaining mass of the system.
To pinpoint the cart's acceleration right before switching the water ejection, we use the thrust calculated earlier and divide it by the remaining mass of the cart.

• Acceleration is expressed as: \[\text{Acceleration} = \frac{\text{Force}}{\text{Remaining mass}}\]
• This calculation explains how the cart's velocity changes per unit time.
• Understanding acceleration is key to predicting the cart's behavior at any stage of its movement.

The acceleration value will give the needed information to understand how the cart behaves as force is continuously applied during the pressure washer's operation.