At an accident scene on a level road, investigators measure a car's skid mark to be 98 m long. It was a rainy day and the coefficient of friction was estimated to be 0.38. Use these data to determine the speed of the car when the driver slammed on (and locked) the brakes.
Well, friction force is mg*mu
That has to equal m*a
a=g*mu
Vf^2=0=Vi^2-2*a*d
solve for Vi
To determine the speed of the car when the driver slammed on and locked the brakes, you can use the principles of kinetic friction and motion equations.
First, let's review how to calculate the force of friction. The force of friction can be calculated using the equation:
Force of friction = coefficient of friction × normal force
In this case, the normal force is equal to the weight of the car, which can be calculated using the equation:
Normal force = mass × gravitational acceleration
Next, we need to determine the deceleration of the car. The deceleration is the rate at which the car slows down, and it can be calculated using the following equation:
Deceleration = Force of friction / mass
Since the force of friction is in the opposite direction of motion, the deceleration will be a negative value.
Now, we can use the equations of motion to calculate the speed of the car. The equation that relates the initial speed, final speed, deceleration, and distance is:
(vf)^2 = (vi)^2 + 2ad
Where:
- vf is the final speed (0 m/s because the car comes to a stop),
- vi is the initial speed (what we want to find),
- a is the deceleration, and
- d is the distance (the length of the skid mark, which is 98 m).
Rearranging the equation, we have:
(vi)^2 = (vf)^2 - 2ad
Since the final speed is 0 m/s, we can simplify to:
(vi)^2 = - 2ad
Now, we can substitute the known values into the equation and solve for vi:
(vi)^2 = - 2 × deceleration × distance
(vi)^2 = - 2 × (Force of friction / mass) × distance
(vi)^2 = - 2 × (coefficient of friction × normal force / mass) × distance
(vi)^2 = - 2 × (coefficient of friction × (mass × gravitational acceleration) / mass) × distance
(vi)^2 = - 2 × coefficient of friction × gravitational acceleration × distance
Finally, we can calculate the initial speed (vi) by taking the square root of both sides of the equation:
vi = square root [ - 2 × coefficient of friction × gravitational acceleration × distance ]
Now, let's substitute the given values:
vi = square root [ - 2 × 0.38 × 9.8 m/s^2 × 98 m ]
Simplifying further:
vi = square root [ - 748.04 ]
Since velocity cannot be negative in this context, we can ignore the negative sign. Therefore:
vi ≈ square root (748.04)
Using a calculator, we find:
vi ≈ 27.34 m/s
So, the approximate speed of the car when the driver slammed on the brakes is 27.34 m/s.