Problem 14 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Police investigators, examining the scene of an accident involving two cars, measure 72 -m-long skid marks of one of the cars, which nearly came to a stop before colliding. The coefficient of kinetic friction between rubber and the pavement is about \(0.80 .\) Estimate the initial speed of that car assuming a level road.

Step-by-Step Solution

Answer

The initial speed was approximately 33.6 m/s.

Step 1: Understanding the Problem

We need to estimate the initial speed of a car based on the length of its skid marks and the coefficient of kinetic friction between the tires and the road. We can use the physics involved in stopping motion to find the initial speed.

Step 2: Identify the Key Equation

To calculate the initial speed, we will use the work-energy principle. The work done by friction equals the change in kinetic energy. The equation is: \[ \frac{1}{2} m v^2 = \mu_k \cdot m \cdot g \cdot d \]where \(v\) is the initial speed, \(\mu_k\) is the coefficient of friction, \(g\) is the acceleration due to gravity, and \(d\) is the distance of the skid marks.

Step 3: Simplify the Equation

Notice that the mass \(m\) cancels out from both sides of the equation, simplifying it to:\[ \frac{1}{2} v^2 = \mu_k \cdot g \cdot d \]

Step 4: Solve for Initial Speed

Rearrange the equation to solve for the initial speed \(v\): \[ v = \sqrt{2 \cdot \mu_k \cdot g \cdot d} \]Now plug in the given values \(\mu_k = 0.80, \ g = 9.8 \ m/s^2, \ d = 72 \ m\).

Step 5: Calculate the Initial Speed

Substitute the given values into the equation:\[ v = \sqrt{2 \cdot 0.80 \cdot 9.8 \cdot 72} \]\[ v = \sqrt{1128.96} \]\[ v \approx 33.6 \ m/s \]

Key Concepts

work-energy principle

The work-energy principle is a fundamental concept in physics that helps us understand how work and energy interact. In this scenario, work is done by kinetic friction to bring the car to a stop. The principle says that the work done by all forces acting on an object equals the change in its kinetic energy.

When the car's tires skid, kinetic friction between the tires and the road converts the car's kinetic energy into heat, slowing it to a stop. The formula used in this situation is: \( \frac{1}{2} m v^2 = \mu_k \cdot m \cdot g \cdot d \) where \(m\) is the mass, \(v\) is the initial speed, \(\mu_k\) is the coefficient of friction, \(g\) is the acceleration due to gravity, and \(d\) is the skid distance. In our case, the mass cancels out, which means every car, regardless of its mass, will have the same initial speed for the same friction and skid length.

This principle allows us to relate the physical properties involved in the stopping motion like friction and the distance of the skid marks, to the car's original speed just before the brakes were applied.

initial speed calculation

Calculating the initial speed of a car from skid marks involves simplifying and using the work-energy principle. Once the mass on both sides of the equation is canceled, the simplified equation becomes: \( \frac{1}{2} v^2 = \mu_k \cdot g \cdot d \).

To find the initial speed \(v\), rearrange the formula: \(v = \sqrt{2 \cdot \mu_k \cdot g \cdot d} \).

This equation tells us that the initial speed depends on three factors:

• The coefficient of kinetic friction, \(\mu_k\) - which measures how much resistance the car faces while skidding.
• The acceleration due to gravity, \(g\) - usually taken as \(9.8 \, \text{m/s}^2\).
• The skid mark length, \(d\) - which is the distance over which the car was sliding.

Substitute the known values (\(\mu_k = 0.80\), \\(g = 9.8 \, \text{m/s}^2\), and \\(d = 72 \, \text{m}\)) into the formula, leading to an estimated initial speed of approximately \(33.6 \, \text{m/s}\).

This result gives a practical estimation, helping law enforcement and accident investigators understand the situation better.

skid marks analysis

Skid marks provide crucial insights into the dynamics of a car accident. By analyzing them, investigators can estimate how fast a car was moving before the driver started braking.

Skid marks are created because of kinetic friction, which acts contrary to the direction of motion when the wheels lock up. The length of the skid marks \(d\) is directly related to the speed because the longer the car skids, the more distance it traveled before stopping.

Using physics, we can determine that even without knowing the car's mass, given a coefficient of kinetic friction and the length of skid marks, we can backtrack and find out how fast the car was moving initially.

In this approach, it's crucial to correctly identify the coefficient of kinetic friction between the tires and the surface, as this directly affects the speed calculation. Additionally, it is important to assume a consistent value for \(g\) as \(9.8 \, \text{m/s}^2\) to ensure accurate calculations.
Ultimately, analyzing skid marks is about looking at physical evidence intelligently. It provides a non-invasive method to piece together an event as critical as an accident.