Problem 14 - Physics for Scientists and Engineers with Modern Physics

Question

(1I) 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 pave- ment is about \(0.80 .\) Estimate the initial speed of that car assuming a level road.

Step-by-Step Solution

Answer

The initial speed of the car was approximately 33.47 m/s.

Step 1: Understanding the Problem

We need to find the initial speed of a car based on the length of the skid marks and the coefficient of kinetic friction. We know the kinetic friction decelerates the car, which nearly stops it from moving. We will use the physics of motion and friction to solve this.

Step 2: Identify the Relevant Formula

The work done by the friction force to bring the car to near halt equals the car's initial kinetic energy. We use the formula: \[ f_k \cdot d = \frac{1}{2} m v^2 \] where \( f_k = \mu_k \cdot m \cdot g \) is the friction force, \( d \) is the distance (72 meters), and \( v \) is the initial velocity.

Step 3: Express Unknowns and Known Values

The coefficient of friction \( \mu_k \) is given as 0.80, the gravitational acceleration \( g \) is approximately 9.81 m/s², and the distance \( d \) is 72 meters. We need to solve for \( v \).

Step 4: Rearrange the Equation and Solve

First express the friction force \( f_k = \mu_k \cdot m \cdot g \). Substitute into the formula:\[ \mu_k \cdot m \cdot g \cdot d = \frac{1}{2} m v^2 \]The mass \( m \) cancels out from both sides, giving:\[ \mu_k \cdot g \cdot d = \frac{1}{2} v^2 \]Re-arrange to find:\[ v^2 = 2 \cdot \mu_k \cdot g \cdot d \] Finally, solve for \( v \):\[ v = \sqrt{2 \cdot \mu_k \cdot g \cdot d} \] Substitute the values:\[ v = \sqrt{2 \cdot 0.80 \cdot 9.81 \cdot 72} \] Calculating gives \( v \approx 33.47 \) m/s.

Key Concepts

Coefficient of Friction

In the world of physics, the coefficient of friction is a vital concept that defines how surfaces interact with each other. It is a dimensionless number that measures the degree of frictional resistance one surface or material provides when it moves over another. When looking at car skid marks, this coefficient helps explain how much force was needed to bring a moving vehicle to a stop or near stop.
In our scenario, the coefficient of friction is given as 0.80, which represents the friction between the car's tires and the road. This means that 80% of the gravitational force's impact contributes to stopping the moving vehicle. Such a value indicates a relatively high friction surface – think of this as the tires having a good grip on the roadway.

Key points to remember:

• The higher the coefficient, the more friction present, resulting in greater stopping power.
• It's crucial for calculating the necessary force to halt the vehicle.
• It varies for different materials and conditions, such as wet or icy roads versus dry roads.

Skid Marks

Skid marks are the black streaks left on a road's surface when the tires of a vehicle are forced to slide instead of roll. They are a visible indicator that tells us the frictional interaction between the road and tire when a vehicle suddenly brakes. These marks are crucial for accident reconstruction analyses, as they help estimate the speed at which a vehicle was traveling before the brakes were applied.
The length of a skid mark correlates with the initial speed and the coefficient of friction at play. Long skid marks, like the 72 meters mentioned in the problem, suggest that significant energy needed to be dissipated to bring the vehicle to a stop.

Important aspects of skid marks:

• Longer skid marks generally imply higher initial speeds.
• They provide clues on the braking force exerted and the road conditions.
• Investigators can use them in formulas together with the friction coefficient to backtrack the velocity.

Work-Energy Principle

The Work-Energy Principle is an essential concept in physics that describes the relationship between work done on an object and its mechanical energy. It asserts that the work done by all forces acting on an object equals the change in its kinetic energy. In our exercise, this principle helps us formulate how the kinetic energy of a moving vehicle changes as friction does work on it, bringing it to a near stop.
The energy the car initially has because of its speed (kinetic energy) is calculated using the formula \( \frac{1}{2} m v^2 \). As the car skids to a halt, the work done by the friction force (\( \mu_k \cdot m \cdot g \cdot d \)) generates heat and causes this kinetic energy to decrease. Thus:

• Work done = change in kinetic energy
• This principle lets us calculate the initial velocity when we know friction and skidding distance.
• The mass of the vehicle cancels out, simplifying the calculations.

By using this principle, we understand that all the car's initial kinetic energy is dissipated by the work of the friction force along the skid marks, ultimately allowing us to determine its initial speed using the given formulas.