Question

A driver's manual states that the stopping distance quadruples as the speed doubles; that is, if it takes \(30 \mathrm{ft}\) to stop a car moving at \(25 \mathrm{mph}\), then it would take \(120 \mathrm{ft}\) to stop a car moving at \(50 \mathrm{mph}\). Justify this statement by using mechanics and the first law of thermodynamics. (Assume that when a car is stopped, its kinetic energy \(\left(\frac{1}{2} m u^{2}\right)\) is totally converted to heat.)

Short answer

The stopping distance quadruples as the speed doubles because the kinetic energy, proportional to stopping distance, quadruples.

Step-by-step solution

Understanding Stopping Distance and Speed

The stopping distance is the distance that a car travels before it comes to a complete stop after the brakes are applied. According to the problem, the stopping distance quadruples as the speed doubles. We need to verify this using the given values of speed and stopping distances. If it takes 30 ft to stop at 25 mph, we expect it to take 120 ft at 50 mph, since 50 mph is double 25 mph.

Kinetic Energy Formula

Kinetic energy is given by the formula \( \frac{1}{2} m u^{2} \), where \( m \) is the mass of the car, and \( u \) is the speed of the car. This represents the energy due to motion. For an object to stop, this kinetic energy must be converted into heat energy through braking.

Doubling Speed and Relationship to Energy

When the speed \( u \) is doubled (from 25 mph to 50 mph), according to the kinetic energy formula, the kinetic energy becomes \( \frac{1}{2} m (2u)^{2} = 2^2 \times \frac{1}{2} m u^{2} = 4 \frac{1}{2} m u^{2} \). This means the kinetic energy is quadrupled when the speed is doubled.

Application to Stopping Distance

Since the stopping involves dissipating all of the kinetic energy into heat, the stopping distance is directly proportional to the kinetic energy. Hence, if kinetic energy quadruples when speed doubles, the stopping distance must also quadruple.

Justification with Given Data

Initially, at 25 mph it takes 30 ft to stop. When the speed doubles to 50 mph, the kinetic energy quadruples, and hence the stopping distance becomes 4 times the initial stopping distance: \( 4 \times 30 \text{ ft} = 120 \text{ ft} \), which aligns with the provided data.

Key concepts

Stopping Distance

The stopping distance of a vehicle is the total distance it travels from the moment the brakes are applied until it comes to a complete stop. It is an important safety consideration when driving, as it affects how quickly a driver can respond to obstacles or emergencies on the road.

• Components of Stopping Distance: Reaction Distance + Braking Distance.
• Reaction Distance: Distance covered from the time the driver perceives the need to stop until the brakes are actually applied.
• Braking Distance: Distance the vehicle travels after the brakes are applied until it stops.

In the given exercise, the primary focus is on braking distance, which depends heavily on the initial speed of the car. According to the exercise, as the speed of the vehicle doubles, the stopping distance quadruples. This is because the kinetic energy, which needs to be dissipated as heat, also quadruples. Understanding stopping distances can greatly enhance driving safety by improving reaction times and awareness of how quickly a car can come to a halt.

Speed and Motion

Speed is a fundamental concept in physics that describes how quickly an object is moving. The relationship between speed and motion is central to studying how objects behave and interact in real-world scenarios.

• Speed Measurement: Speed is typically measured in units of distance per time, such as miles per hour (mph) or meters per second (m/s).
• Acceleration: Can cause changes in speed, described as the rate of change of velocity over time.
• Uniform Motion: Motion at constant speed in a straight line.

In the context of the provided exercise, the speed of the car is crucial in determining the stopping distance. This is because the kinetic energy, which is reliant upon the square of the velocity, dictates how much distance is required to stop the car. Doubling the speed, as shown, significantly increases the kinetic energy and hence the stopping distance. Understanding speed and motion helps in predicting and controlling a vehicle's behavior effectively.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In the context of stopping a car, thermodynamics helps explain how kinetic energy is transformed into heat energy.

• First Law of Thermodynamics: Energy cannot be created or destroyed, only transformed from one form to another.
• Energy Transformation: When a car stops, its kinetic energy is converted to heat energy through the braking system.

According to the first law of thermodynamics, the kinetic energy involved in the motion of the car must be entirely dissipated as heat for the car to come to a stop. This heat is then absorbed by the brakes and released into the environment. This conversion explains why, as the kinetic energy quadruples with speed, the stopping distance also increases. Thermodynamics provides the framework for understanding this energy exchange during braking.

Physics Mechanics

Physics mechanics deals with the motion and forces that act upon objects. It's the foundation for concepts such as kinetic energy and stopping distance.

• Kinetic Energy: Represented by the formula \( \frac{1}{2} m u^{2} \), it is a measure of the energy an object possesses due to motion.
• Newton's Laws: Essential principles that describe motion, including inertia and the influence of forces.
• Force and Motion: Related through the equation \( F = ma \), which involves mass, acceleration, and force.

Understanding mechanics is crucial for explaining how and why a car stops as it does. The kinetic energy increases in proportion to the square of the velocity, meaning that a small increase in speed can result in a large increase in energy that needs to be dissipated. Mechanics allows us to quantify and relate the physical forces and energy transformations that occur when stopping a moving car. The exercise illustrates these principles clearly through the relationship between speed and stopping distance. Mechanics encapsulates real-world applications, from stopping a car to predicting motion patterns.