At an accident scene on a level road, investigators measure a car's skid mark to be 68 m long. It
was a rainy day and the coefficient of friction was estimated to be 0.40. Using the work-energy
theorem determine the speed of the car when the driver slammed on (and locked) the brakes. Why
does the car's mass not matter?
To determine the speed of the car when the driver slammed on the brakes, we can use the work-energy theorem. The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done on the car is equal to the friction force acting on it multiplied by the distance over which the force is applied.
To calculate the speed of the car, we can equate the work done by the friction force to the change in kinetic energy of the car. The work done by the friction force is given by the formula:
Work = Friction Force * Distance
The friction force can be calculated using the equation:
Friction Force = Coefficient of Friction * Normal Force
The normal force is the force exerted by the ground on the car, which is equal to the weight of the car. The weight of an object is given by the formula:
Weight = Mass * Gravity
In this case, the mass of the car does not matter because it will cancel out when calculating the friction force. Therefore, the mass of the car does not affect the final calculation of the car's speed. The only relevant factors are the coefficient of friction and the length of the skid mark.
Once we have the friction force, we can calculate the work done by multiplying it by the distance (skid mark length). Since the work done is equal to the change in kinetic energy, we can equate it to the initial kinetic energy of the car:
Work = Change in Kinetic Energy
The initial kinetic energy of the car is given by the formula:
Initial Kinetic Energy = 0.5 * Mass * Initial Velocity^2
By solving this equation for the initial velocity (speed), we can determine the speed of the car when the driver slammed on the brakes.
Note: It is important to remember that the work-energy theorem assumes no other forces or energy losses during the process. In reality, there may be other factors to consider, such as air resistance or energy dissipation due to deformations in the car, which may slightly affect the final calculated speed.
To determine the speed of the car when the driver slammed on the brakes, we can use the work-energy theorem:
The work-energy theorem states that the work done on an object is equal to its change in kinetic energy. In this case, the work done on the car will be equal to the change in its kinetic energy due to braking.
The work done on the car is given by the equation:
Work = Force * Distance
The force acting on the car during braking is the friction force between the tires and the road. This friction force can be calculated using the equation:
Force = friction coefficient * normal force
In this case, the normal force is equal to the weight of the car, which is given by:
Normal force = mass * gravity
Substituting this into the equation for force, we get:
Force = friction coefficient * mass * gravity
Now, we can substitute the equation for force into the equation for work:
Work = (friction coefficient * mass * gravity) * distance
The work done on the car is equal to the change in its kinetic energy due to braking. Since the car comes to a stop, its final kinetic energy is zero. Therefore, the work done on the car is equal to its initial kinetic energy.
The initial kinetic energy of the car is given by the equation:
Initial Kinetic Energy = (1/2) * mass * velocity^2
Setting the work done equal to the initial kinetic energy and rearranging the equation, we can solve for the initial velocity:
(friction coefficient * mass * gravity * distance) = (1/2) * mass * velocity^2
Cancelling out the mass on both sides of the equation, we get:
friction coefficient * gravity * distance = (1/2) * velocity^2
Now we solve for velocity:
velocity = sqrt(2 * friction coefficient * gravity * distance)
The mass of the car cancels out in this equation, so it does not affect the final answer. The speed of the car when the driver slammed on the brakes can be calculated using the given values for the coefficient of friction, gravity, and the length of the skid mark.