Question

To contain some unruly demonstrators, the riot squad approaches with fire hoses. Suppose that the rate of flow of water through a fire hose is \(24 \mathrm{kg} / \mathrm{s}\) and the stream of water from the hose moves at \(17 \mathrm{m} / \mathrm{s}\). What force is exerted by such a stream on a person in the crowd? Assume that the water comes to a dead stop against the demonstrator's chest.

Short answer

The force exerted by the water stream on a person is 408 N in the opposite direction of the flow.

Step-by-step solution

Understand the Problem Concept

To find the force exerted by the stream of water on a person, we need to understand that the force can be calculated using the momentum change per unit time. This force is given by the equation: \[ F = \frac{{\Delta p}}{{\Delta t}} \] where \( \Delta p \) is the change in momentum, and \( \Delta t \) is the time interval.

Calculate the Initial Momentum of Water

The initial momentum \( p_1 \) of the water can be calculated using the formula: \[ p_1 = mv \] where \( m \) is the mass flow rate and \( v \) is the velocity. Given that the mass flow rate is \( 24 \text{ kg/s} \) and the velocity is \( 17 \text{ m/s} \), we have:\[ p_1 = 24 \times 17 = 408 \text{ kg m/s} \] per second.

Describe the Change in Momentum

Since the water comes to a complete stop against the person's chest, the final momentum \( p_2 \) is zero. The change in momentum \( \Delta p \) is therefore:\[ \Delta p = p_2 - p_1 = 0 - 408 \text{ kg m/s} = -408 \text{ kg m/s} \] per second.

Calculate the Force Exerted

According to the relation for force due to momentum change, we use the formula:\[ F = \frac{{\Delta p}}{{\Delta t}} \]Given that \( \Delta p = -408 \text{ kg m/s} \) per second (since it happens continuously), the force \( F \) is:\[ F = -408 \text{ N} \]The negative sign indicates the direction of the force is opposite to the direction of the initial motion, which means the force is exerted on the demonstrator in the opposite direction of the water flow.

Key concepts

Force Calculation

Calculating force involves understanding the relationship between momentum and time. To determine the force exerted by a stream of water from a fire hose, we employ the equation \( F = \frac{\Delta p}{\Delta t} \).
This formula is derived from Newton's second law, which connects force, mass, and acceleration. In our context, the change in momentum per unit time provides the rate of change in momentum, which is equivalent to the force. Given this relationship, we can deduce how an action, like stopping a stream of water, results in a force on the object it encounters.

Mass Flow Rate

Mass flow rate is a key concept in fluid dynamics. It describes the mass of fluid passing through a given surface per unit time. In the problem, this is specified as \( 24 \text{ kg/s} \).
The constant flow of water at this rate signifies a continuous stream, which can be crucial in understanding force calculations. For water and demonstrators in our scenario, this steady rate allows us to precisely calculate how much mass impacts the demonstrator's chest per second, contributing directly to the momentum change.

Velocity

Velocity is not just the speed of an object but also the direction of its motion. In this context, the water moving at \( 17 \text{ m/s} \) towards the demonstrator tells us how fast and in what direction the water is traveling.
Velocity affects momentum because momentum is the product of mass and velocity: \( p = mv \). Thus, knowing the velocity helps estimate the initial momentum of the water stream before it hits the demonstrator, guiding us in the overall solution of force exertion.

Momentum Change

Momentum change occurs when the velocity of a mass changes. This principle is central to calculating forces that result from such changes.
Initially, the water's momentum is calculated using \( p_1 = mv \), resulting in \( 408 \text{ kg m/s} \). When the water comes to a stop, its momentum becomes zero, thus creating a change or delta in momentum: \( \Delta p = 0 - 408 \text{ kg m/s} \).
The direction is crucial too. The negative sign in our calculations means the force is acting in the opposite direction of the water's original flow - specifically, on the demonstrator. Understanding momentum change allows us to see how forces physically manifest in real-world scenarios.