Problem 31 - University Physics with Modern Physics

Question

A car is traveling on a level road with speed \(v_0\) at the instant when the brakes lock, so that the tires slide rather than roll. (a) Use the work energy theorem to calculate the minimum stopping distance of the car in terms of \(v_0\), \(g\), and the coefficient of kinetic friction \(\mu_k\) between the tires and the road. (b) By what factor would the minimum stopping distance change if (i) the coefficient of kinetic friction were doubled, or (ii) the initial speed were doubled, or (iii) both the coefficient of kinetic friction and the initial speed were doubled?

Step-by-Step Solution

Answer

(a) \(d = \frac{v_0^2}{2\mu_k g}\); (b) (i) half, (ii) twice, (iii) same.

Step 1: Understanding the scenario

Initially, the car is moving with speed \(v_0\) on a level road. When brakes are locked, the tires start sliding, bringing the car to a stop due to kinetic friction. Our task is to find the minimum stopping distance using the work-energy theorem.

Step 2: Applying the Work-Energy Theorem

The work-energy theorem states that the work done on an object is equal to its change in kinetic energy. For this problem, the initial kinetic energy is \(\frac{1}{2} m v_0^2\), and the final kinetic energy is 0 (since the car stops). The work done by friction is \(-f_k \cdot d\), where \(f_k = \mu_k m g\) is the force of friction and \(d\) is the stopping distance.

Step 3: Setting Up the Equation

According to the work-energy theorem, the work done by friction equals the change in kinetic energy: \[-f_k \cdot d = 0 - \frac{1}{2} m v_0^2\] Rearrange to get: \[f_k \cdot d = \frac{1}{2} m v_0^2\]

Step 4: Solving for Stopping Distance

Substitute \(f_k = \mu_k m g\) into the equation:\[\mu_k m g \cdot d = \frac{1}{2} m v_0^2\]Cancel \(m\) from both sides:\[\mu_k g \cdot d = \frac{1}{2} v_0^2\]Solve for \(d\):\[d = \frac{v_0^2}{2\mu_k g}\]

Step 5: Analyzing Changes in Scenario

(i) If \(\mu_k\) is doubled, \(d = \frac{v_0^2}{2(2\mu_k) g} = \frac{v_0^2}{4\mu_k g}\), so \(d\) becomes half its original value.(ii) If \(v_0\) is doubled, \(d = \frac{(2v_0)^2}{2\mu_k g} = \frac{4v_0^2}{2\mu_k g} = 2\frac{v_0^2}{2\mu_k g}\), so \(d\) becomes twice its original value.(iii) If both \(\mu_k\) and \(v_0\) are doubled, \(d = \frac{(2v_0)^2}{2(2\mu_k) g} = \frac{4v_0^2}{4\mu_k g} = \frac{v_0^2}{2\mu_k g}\), so \(d\) remains the same.

Key Concepts

Stopping Distance

Stopping distance is an important concept in understanding vehicle dynamics and safety. Simply put, it's the distance a vehicle travels between the point at which the brakes are applied and when the vehicle comes to a stop. In mathematical terms, it is calculated using the work-energy theorem, an essential principle that relates work done by forces to changes in kinetic energy.
The minimum stopping distance is particularly relevant for assessing braking performance under ideal conditions. For a car initially moving with speed \(v_0\), the stopping distance \(d\) can be derived by examining the energy transformations when brakes lock the wheels. The initial kinetic energy given by \(\frac{1}{2} m v_0^2\) entirely converts into work done by frictional force to stop the vehicle.
Using the work-energy theorem, we calculate the stopping distance \(d\) as:

• \(d = \frac{v_0^2}{2\mu_k g}\)

Where \(\mu_k\) is the coefficient of kinetic friction, and \(g\) is the acceleration due to gravity. Key scenarios affecting \(d\) include changes in \(v_0\) and \(\mu_k\), illustrating why these variables matter in vehicle braking.

Coefficient of Kinetic Friction

The coefficient of kinetic friction \(\mu_k\) is a measure of how easily one surface moves over another when friction is present. It plays a crucial role when analyzing the stopping distance of a vehicle.
The coefficient is a dimensionless quantity representing the frictional resistance relative to the normal force between the tires and the road. In vehicles, knowing \(\mu_k\) helps predict the stopping performance. A higher value generally indicates a rougher surface or better grip, leading to faster stops. Conversely, a lower \(\mu_k\) suggests a smoother surface, like ice, causing longer stopping distances.
In the formula for stopping distance \(d = \frac{v_0^2}{2\mu_k g}\), \(\mu_k\) directly influences the outcome:

• Doubling \(\mu_k\) reduces \(d\) by half, indicating how improved tire conditions or road quality enhance stopping power.

Understanding \(\mu_k\) is vital for automotive engineers and safety planners to ensure effective brake system design and road safety.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. For a car moving at speed \(v_0\), the kinetic energy is expressed as:

• \(KE = \frac{1}{2} m v_0^2\)

Here, \(m\) represents the mass of the vehicle, and \(v_0\) is its velocity. When exploring braking scenarios, understanding kinetic energy provides insights into why faster cars or heavier vehicles need longer distances to stop.
The work-energy theorem seamlessly links kinetic energy to work done by friction to stop the car. As kinetic energy transitions to other forms (like heat during braking), it directly quantifies the necessary energy dissipation for complete stops.
When initial speed \(v_0\) doubles, the kinetic energy quadruples, since energy is proportional to the velocity squared. This relationship underscores the exponential increase in required stopping distance, making speed a critical factor in road safety:

• If \(v_0\) is doubled, stopping distance \(d\) doubles.

Grasping the concept of kinetic energy is essential to understanding vehicle dynamics and ensuring safe driving practices.