A police officer is called in for accident reconstruction. He measures 11-meter long skid marks and he knows that for that particular tire/pavement combination, the coefficient of kinetic friction is 0.65. He also knows that the car was traveling at 3.5 m/s just before impact (the end of the skid marks). What was the speed of the car before it began the skid? Convert this to miles per hour. (Note: 1 mile = 1609 meters).
The correct answer and explanation is :
Step 1: Understanding the Problem
A police officer is reconstructing an accident and measures that a car left 11-meter-long skid marks before stopping. The coefficient of kinetic friction between the tires and the pavement is 0.65. The car’s speed at the end of the skid (just before impact) was 3.5 m/s. Our goal is to determine the car’s speed before the skid began and convert it to miles per hour (mph).
Step 2: Applying the Work-Energy Principle
The work-energy theorem states that the work done by friction equals the change in kinetic energy:
[
W = \Delta KE
]
Since friction is the only force doing work, the equation becomes:
[
\frac{1}{2} m v_i^2 – \frac{1}{2} m v_f^2 = – f_k d
]
Where:
- ( v_i ) = initial speed before skidding (m/s) (to be found)
- ( v_f = 3.5 ) m/s (final speed before impact)
- ( d = 11 ) m (skid length)
- ( f_k = \mu_k mg ) (friction force)
- ( \mu_k = 0.65 ) (coefficient of kinetic friction)
- ( g = 9.81 ) m/s² (acceleration due to gravity)
Since ( f_k = \mu_k mg ), and work done by friction is:
[
W = -(\mu_k m g) d
]
Cancelling mass ( m ) from both sides:
[
\frac{1}{2} v_i^2 – \frac{1}{2} v_f^2 = -\mu_k g d
]
Rearranging:
[
v_i^2 = v_f^2 + 2 \mu_k g d
]
Substituting the values:
[
v_i^2 = (3.5)^2 + 2(0.65)(9.81)(11)
]
[
v_i^2 = 12.25 + 2(0.65)(9.81)(11)
]
[
v_i^2 = 12.25 + 140.51
]
[
v_i^2 = 152.76
]
[
v_i = \sqrt{152.76}
]
[
v_i \approx 12.36 \text{ m/s}
]
Step 3: Convert to Miles per Hour (mph)
Using the conversion:
[
1 \text{ m/s} = 2.237 \text{ mph}
]
[
v_i \approx 12.36 \times 2.237
]
[
v_i \approx 27.66 \text{ mph}
]
Final Answer:
The car was traveling about 27.66 mph before it began skidding.
Explanation:
When a vehicle begins skidding, its kinetic energy is gradually dissipated by the friction force acting between the tires and the road. This force is proportional to the normal force (which is simply the weight of the car in this case) and the coefficient of kinetic friction.
By using the work-energy principle, we calculated how much energy was lost during the skid and used it to determine the initial speed. The final speed just before impact (3.5 m/s) was included in the calculation to ensure accuracy.
This method is commonly used by accident reconstruction experts to estimate pre-impact speeds of vehicles based on physical evidence like skid marks, road conditions, and vehicle dynamics.