Question

A \(65-\mathrm{kg}\) bicyclist rides his \(8.8-\mathrm{kg}\) bicycle with a speed of \(14 \mathrm{~m} / \mathrm{s}\). (a) How much work must be done by the brakes to bring the bike and rider to a stop? (b) What is the magnitude of the braking force if the bicycle comes to rest in \(3.5 \mathrm{~m}\) ?

Short answer

(a) 7227.6 J of work is required; (b) Braking force is 2065 N.

Step-by-step solution

Combine Masses

First, we need to find the total mass of the bicyclist and the bicycle. Add the mass of the bicyclist (65 kg) to the mass of the bicycle (8.8 kg):\[m_{total} = 65 \, \text{kg} + 8.8 \, \text{kg} = 73.8 \, \text{kg}\]

Calculate Initial Kinetic Energy

Next, calculate the initial kinetic energy of the system using the formula for kinetic energy: \[KE_{initial} = \frac{1}{2} m_{total} v^2\]Substitute in the values:\[KE_{initial} = \frac{1}{2} \times 73.8 \, \text{kg} \times (14 \, \text{m/s})^2 = 7227.6 \, \text{J}\]

Determine Work Done by Brakes

The work done by the brakes is equal to the initial kinetic energy, since it is required to bring the bicycle to a stop (final kinetic energy is zero):\[W = KE_{initial} = 7227.6 \, \text{J}\]This is the work that must be done by the brakes.

Calculate Braking Force

To find the braking force, we use the work-energy principle where work is also given as force multiplied by distance:\[W = F \cdot d\]Solve for the force:\[F = \frac{W}{d} = \frac{7227.6 \, \text{J}}{3.5 \, \text{m}} = 2065.03 \, \text{N}\]Thus, the magnitude of the braking force is approximately 2065 N.

Key concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It depends on both the mass of the object and its velocity. In our exercise, we start by understanding how kinetic energy plays a role in bringing the bicycle to a halt.
To calculate kinetic energy, we use the formula:

• \( KE = \frac{1}{2} mv^2 \)

Here, \( m \) is the mass of the object (in our case, the combined mass of the bicyclist and bicycle), and \( v \) is the velocity. Substituting the values, we find that the initial kinetic energy of the bicyclist and bicycle is \( 7227.6 \) Joules. This energy represents the work needed to stop the bicycle.
The concept here is simple: whatever is in motion, stores energy, which can be observed through its kinetic energy. Reducing this energy to zero would mean bringing the moving bicycle to a stop.

Braking Force

When a bicycle comes to rest, the brakes apply a force in the direction opposite to the motion. This is known as the braking force.
To find out how strong the braking force needs to be, we leverage the work-energy principle, which relates work done by a force to the change in kinetic energy.

Work-Energy Principle

According to the work-energy principle, the work done by the braking force is equal to the change in kinetic energy. Since the bike comes to a complete stop, its final kinetic energy is zero, which means the entire initial kinetic energy is used up by the brakes.
In our calculation, the work done \( W \), already found as \( 7227.6 \) Joules, equates to the force \( F \) applied over a displacement \( d \) (distance).

• \( W = F \cdot d \)

By rearranging this formula to solve for the force, we determine that the braking force \( F \) is approximately 2065 Newtons. This force is what decelerates the bicycle, bringing it to a rest over a distance of \( 3.5 \) meters.

Mechanical Work

Mechanical work, in simple terms, refers to the process of energy transfer via a force acting upon an object over a distance. In the case of the bicycle, the work done by the brakes is crucial for stopping it.
The concept of mechanical work is represented by the equation:

• \( W = F \cdot d \)

Here, \( W \) denotes work, \( F \) is the force applied (in our case, the braking force), and \( d \) refers to the distance over which the force is applied. With the help of this formula, you can calculate how much energy is transferred to halt the movement of any object.
In our problem, knowing the work done by the brakes (\( 7227.6 \) Joules) and the distance (\( 3.5 \) meters), we are able to find the magnitude of the braking force. This aspect umanufactors our understanding of how much effort (or mechanical work) it takes to bring a moving bicycle to a complete stop.