Question

A 1100-kg car is coasting on a horizontal road with a speed of \(19 \mathrm{~m} / \mathrm{s}\). After passing over an unpaved, sandy stretch \(32 \mathrm{~m}\) long, the car's speed has decreased to \(12 \mathrm{~m} / \mathrm{s}\). (a) Was the net work done on the car positive, negative, or zero? Explain. (b) Find the magnitude of the average net force on the car in the sandy section of the road.

Short answer

(a) The net work done is negative. (b) The magnitude of the average net force on the car is 3739.06 N (acting opposite to motion).

Step-by-step solution

Identify the Problem

We need to determine the effect of the net work done on the car and the average net force acting on it while it moves across a 32-meter sandy stretch. We are given the initial and final velocities, the car's mass, and the distance of the sandy stretch.

Apply Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. Calculate the initial and final kinetic energies, and then find the net work done. The formula for kinetic energy is given by \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity.

Calculate Initial and Final Kinetic Energies

Calculate the initial kinetic energy: \( KE_{initial} = \frac{1}{2} \times 1100 \times 19^2 = 198550 \, \text{J} \). Calculate the final kinetic energy: \( KE_{final} = \frac{1}{2} \times 1100 \times 12^2 = 79200 \, \text{J} \).

Determine the Net Work Done

The net work done on the car is the change in kinetic energy, which is \( KE_{final} - KE_{initial} \). Substitute the values: \( ext{Net Work} = 79200 - 198550 = -119350 \, \text{J} \). Since the net work is negative, it indicates that the energy is lost, likely due to friction in the sandy stretch.

Determine the Nature of Work

Since the net work done is negative, it means that the net work on the car is negative. This indicates that external forces (like friction) are doing work in the opposite direction of the car's motion, slowing it down.

Calculate Average Net Force

The average net force can be calculated using the work-energy principle where \( W = F_{avg} \cdot d \). Rearrange to find \( F_{avg} = \frac{W}{d} \). Substitute the values from Step 4 and distance \( d = 32 \, \text{m} \): \( F_{avg} = \frac{-119350}{32} = -3739.06 \, \text{N} \).

Interpret the Force Result

The negative sign of the average net force indicates that the force acting on the car is in the opposite direction of its motion, consistent with our earlier conclusion that the work done was negative.

Key concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It depends on two factors: the mass of the object and its velocity. The formula for calculating kinetic energy is \( KE = \frac{1}{2} m v^2 \),where:

• \( m \) represents the mass of the object,
• \( v \) is the velocity of the object.

For example, if a car has a larger velocity or mass, it has more kinetic energy. In our exercise, the initial kinetic energy of the car was 198550 Joules due to its initial velocity of 19 m/s. After traveling through the sandy stretch and slowing to 12 m/s, its kinetic energy decreased to 79200 Joules. This change vividly highlights the loss of kinetic energy, which is crucial for understanding the Work-Energy Theorem.

Understanding this concept helps us see how energy transforms and is transferred in different scenarios.

Net Work

Net work refers to the total work done by all forces acting on an object. It can lead to either an increase or decrease in an object's kinetic energy. According to the Work-Energy Theorem, the net work done is equal to the change in kinetic energy, given by \( W_{net} = KE_{final} - KE_{initial} \). This theorem helps explain how energy is moved or absorbed by objects.

The net work is positive if an object's speed increases, indicating energy gain. Conversely, if the speed decreases, the net work is negative, signaling an energy loss. In our scenario, as the car slows down when passing over the sandy stretch, the net work is -119350 Joules. Such negative net work suggests that a force (likely friction from the sand) acts against the car's motion, draining its kinetic energy.

Average Net Force

When discussing the average net force, we refer to the constant force that, if applied, would produce the same work as the actual variable forces over a given distance. It's calculated using the formula: \( F_{avg} = \frac{W_{net}}{d} \), where:

• \( W_{net} \) is the net work done,
• \( d \) is the distance over which the force acts.

In this exercise, the sandy stretch of the road is 32 meters long. With a net work of -119350 Joules, the average net force is calculated to be -3739.06 Newtons.

The negative sign here indicates that this force is acting in the opposite direction of the car's motion. This aligns with our understanding of the work done and shows how forces can impact an object's speed and motion direction.