A police officer investigating an accident estimates that a moving car hit a stationary car at 24km/h .
If the moving car left skid marks 45m long, and if the coefficient of kinetic friction is 0.71, what was the initial speed of the moving car?
Express your answer using two significant figures.
745
To determine the initial speed of the moving car, we need to use the equations of motion and the concept of friction.
First, let's consider the equation of motion for a moving car under constant acceleration:
v² = u² + 2as
Where:
- v is the final velocity (0 km/h since the car came to a stop)
- u is the initial velocity (what we're trying to find)
- a is the acceleration
- s is the distance (length of the skid mark)
Now, let's calculate the acceleration of the car using the concept of friction:
Friction force = coefficient of friction * normal force
The normal force is the force exerted by the stationary car on the moving car, which is equal to the weight of the moving car.
Weight = mass * gravity
Assuming the mass of the car is m and the acceleration due to gravity is g, the normal force will be m * g.
Using Newton's second law (F = ma), the friction force will be:
Friction force = m * g * coefficient of friction
Friction force = (m * g) * 0.71
Now, we can use the definition of frictional force to determine the acceleration:
Friction force = mass * acceleration
(m * g) * 0.71 = m * a
0.71 * g = a
Finally, substituting the values into the equation of motion:
0 = u² + 2 * 0.71 * g * s
Solving for u:
u² = - 2 * 0.71 * g * s
Taking the square root of both sides, we find:
u = √(-2 * 0.71 * g * s)
Now, we can substitute the given values:
g = 9.8 m/s² (acceleration due to gravity)
s = 45 m
u = √(-2 * 0.71 * 9.8 * 45)
Calculating this expression reveals:
u ≈ 19.3 m/s
Converting to km/h:
u ≈ 19.3 m/s * (3.6 km/h / 1 m/s) ≈ 69.5 km/h
Therefore, the initial speed of the moving car was approximately 69.5 km/h (rounded to two significant figures).