Problem 86 - University Physics with Modern Physics

Question

You are called as an expert witness in the trial of a traffic violation. The facts are these: A driver slammed on his brakes and came to a stop with constant acceleration. Measurements of his tires and the skid marks on the pavement indicate that he locked his car's wheels, the car traveled 192 ft before stopping, and the coefficient of kinetic friction between the road and his tires was \(0.750 .\) The charge is that he was speeding in a \(45-\mathrm{mi} / \mathrm{h}\) zone. He pleads innocent. What is your conclusion, guilty or innocent? How fast was he going when he hit his brakes?

Step-by-Step Solution

Answer

Guilty. He was initially going 52.30 mi/h, exceeding the 45 mi/h speed limit.

Step 1: Identify the Known Values

The problem provides the following known values: the distance (d) the car traveled while skidding is 192 ft, the coefficient of kinetic friction (\(\mu_k\)) is 0.750, and the acceleration due to gravity (g) is approximately 32.2 \(\text{ft/s}^2\). The speed limit is 45 mi/h.

Step 2: Calculate the Frictional Force

The frictional force \(F_f\) can be found using \(F_f = \mu_k \times m \times g\). Since we only need the acceleration, we use \(a = \mu_k \times g\). Substituting the known values, we get:\[ a = 0.750 \times 32.2 \approx 24.15 \text{ ft/s}^2 \].

Step 3: Use Kinematics for Initial Velocity

We know the final velocity \(v_f\) is 0 (because the car stopped), and we want to find the initial velocity \(v_i\). We can use the kinematic equation: \[ v_f^2 = v_i^2 + 2ad \]Substituting the known values:\[ 0 = v_i^2 + 2(-24.15)(192) \]Solve for \(v_i\):\[ v_i^2 = 2 \times 24.15 \times 192 \]\[ v_i = \sqrt{2 \times 24.15 \times 192} \approx 76.74 \text{ ft/s} \].

Step 4: Convert Initial Velocity to Miles per Hour

Convert the velocity from ft/s to mi/h using the conversion factor (1 mi/h = 1.467 ft/s):\[ v_i = \frac{76.74 \text{ ft/s}}{1.467} \approx 52.30 \text{ mi/h} \].

Step 5: Compare with the Speed Limit

The calculated initial speed \(52.30 \text{ mi/h}\) is greater than the speed limit of \(45 \text{ mi/h}\). This implies the driver was speeding at the time he began braking.

Key Concepts

kinematic equations

In the realm of physics problem solving, kinematic equations are vital tools for analyzing the motion of objects. These equations relate the details of motion such as initial velocity, final velocity, acceleration, time, and displacement. In the given problem, the driver comes to a complete stop, which clearly indicates a change in velocity. We use the kinematic equation: \[ v_f^2 = v_i^2 + 2ad \] Here, \(v_f\) is the final velocity, \(v_i\) is the initial velocity, \(a\) is the acceleration, and \(d\) is the displacement. Since the car stopped, \(v_f = 0\). This allows us to rearrange the equation to find the initial velocity \(v_i\). The equation simplifies to: \[ 0 = v_i^2 + 2(-a)(d) \] Solving for \(v_i\), we get: \[ v_i^2 = 2ad \] This approach helps us determine how fast the car was going before the brakes were applied. Kinematic equations are powerful because they provide a way to calculate unknown variables given some initial data, without the need for complex measurements or assumptions.

frictional force calculation

The frictional force calculation is essential for understanding how objects slow down when subjected to a resistive force. When a car skids, the friction between the tires and the road dissipates the kinetic energy. The frictional force \(F_f\) is given by the formula: \[ F_f = \mu_k \times m \times g \] However, in this scenario, we only need the frictional acceleration to solve the problem. Thus, we use the equation: \[ a = \mu_k \times g \] Where \(\mu_k\) is the coefficient of kinetic friction and \(g\) is the acceleration due to gravity. Substituting the known values, we find: \[ a = 0.750 \times 32.2 \text{ ft/s}^2 \approx 24.15 \text{ ft/s}^2 \] This acceleration represents the deceleration caused by friction, which will be used in the kinematic equation to calculate the initial speed of the vehicle. Understanding how friction affects motion is crucial for analyzing problems involving braking and stopping distances.

unit conversion in physics

Unit conversion in physics ensures that measurements are consistent and calculations are accurate. In this specific exercise, we need to convert the initial velocity from feet per second (ft/s) to miles per hour (mi/h) for comparison with the speed limit. The conversion factor between these units is: \[ 1 \text{ mi/h} = 1.467 \text{ ft/s} \] By dividing the velocity in \(\text{ft/s}\) by the conversion factor, we obtain the speed in \(\text{mi/h}\): \[ v_i = \frac{76.74 \text{ ft/s}}{1.467} \approx 52.30 \text{ mi/h} \] Accurate unit conversion is a crucial step because using the wrong units can lead to incorrect conclusions, as was important in determining whether the driver exceeded the speed limit. Keeping track of units throughout a problem helps maintain clarity and precision in calculations.