Problem 85 - College Physics

Question

You are in charge of an accident reconstruction case for the local police department. In order to determine car speeds, skid mark lengths are measured. To determine the coefficient of kinetic friction, you get into an identical car, and at a speed of \(65.2 \mathrm{mi} / \mathrm{h}\), you lock its brakes and skid \(51.5 \mathrm{~m}\) to rest. (a) Determine the car's deceleration. (b) What is the coefficient of kinetic friction between the tires and road surface? (c) The car in the accident actually skidded \(57.3 \mathrm{~m}\). What was its initial speed?

Step-by-Step Solution

Answer

(a) Deceleration is \(-8.25 \text{ m/s}^2\). (b) Coefficient of friction is \(0.84\). (c) Initial speed was \(68.8 \text{ mi/h}\).

Step 1: Convert Speed to Meters Per Second

First, convert the car's initial speed from miles per hour to meters per second. We use the conversion: \(1 \text{ mile} = 1609.34 \text{ meters}\) and \(1 \text{ hour} = 3600 \text{ seconds}\). Thus, \(65.2 \text{ mi/h} = 65.2 \times \frac{1609.34}{3600} \text{ m/s} = 29.14 \text{ m/s}\).

Step 2: Determine the Car's Deceleration

Using the kinematic equation \(v^2 = u^2 + 2a s\), where \(v = 0\) (final velocity), \(u = 29.14 \text{ m/s}\) (initial velocity), \(s = 51.5 \text{ m}\) (distance), solve for \(a\) (deceleration): \[0 = (29.14)^2 + 2a(51.5)\] \[a = -\frac{(29.14)^2}{2 \times 51.5} \approx -8.25 \text{ m/s}^2\]

Step 3: Calculate Coefficient of Kinetic Friction

Use the formula for deceleration due to friction: \(a = \mu g\), where \(\mu\) is the coefficient of kinetic friction and \(g = 9.81 \text{ m/s}^2\) (acceleration due to gravity). Solve for \(\mu\):\[\mu = \frac{a}{g} = \frac{-8.25}{9.81} \approx 0.84\]

Step 4: Determine Initial Speed of Accident Car

Using the same kinematic formula \(v^2 = u^2 + 2a s\), now calculate the initial speed \(u\) for the accident car that skidded \(57.3 \text{ m}\) with deceleration \(a = -8.25 \text{ m/s}^2\): \[v^2 = 0, \quad 0 = u^2 + 2(-8.25)(57.3)\] \[u^2 = 2 \times 8.25 \times 57.3 \approx 947.745\] \[u = \sqrt{947.745} \approx 30.78 \text{ m/s}\]

Step 5: Convert Initial Speed to Miles Per Hour

Convert the calculated speed of the accident car from meters per second back to miles per hour: \[30.78 \text{ m/s} = 30.78 \times \frac{3600}{1609.34} \text{ mi/h} \approx 68.8 \text{ mi/h}\]

Key Concepts

Accident Reconstruction

Accident reconstruction is a crucial process in understanding what happened during a vehicle collision. It involves collecting data like skid marks and using physics principles to deduce speeds and forces involved. The length of skid marks can provide insights into how brakes were used and the speed of the vehicle at the time of braking. By analyzing these physical traces, experts can determine if excess speed was a factor in an accident.
This is essential for assessing liability in accidents and improving road safety measures. When conducting accident reconstruction, investigators often reenact scenarios in controlled environments to obtain accurate measurements and compare them to the actual incidents. This helps in validating their findings and ensuring correctness.

Car Deceleration

Deceleration is a decrease in speed over time, crucial in understanding braking events. In physics, it's treated as a negative acceleration. In our example, deceleration occurs when the car's brakes are locked, causing it to skid.
The kinematic equation used to determine deceleration is:

• \(v^2 = u^2 + 2as\)

'v' represents the final velocity (0 for complete stops), 'u' is the initial velocity, 'a' is acceleration (negative for deceleration), and 's' is the distance over which the deceleration occurs. By rearranging this equation, we can solve for 'a'. The negative result indicates deceleration (\(-8.25 \text{ m/s}^2\) in this case).
Understanding deceleration helps reconstruct accidents and determine whether the car was driving within safe limits before the incident.

Initial Speed Calculation

Calculating a car's initial speed during an incident involves using physics principles and data observed from the accident scene, such as skid mark lengths. This data is plugged into kinematic equations to extract information about initial velocities before accidents.
In our example, knowing the deceleration and the distance the car skidded (from the accident site), we apply the same kinematic formula used for deceleration:

• \(v^2 = u^2 + 2as\)

Plug in the data: if the final speed \(v\) is 0, 'u' (initial speed) can be solved by rearranging the formula. For the accident car that skidded further than the test (57.3 m compared to 51.5 m), the initial speed is calculated to be \(30.78 \, \text{m/s}\) (about 68.8 mi/h).
This process shows how physics and mathematical calculations can provide clarity about pre-accident conditions, identifying whether excessive speed was a factor.