You are the juror of a case involving a drunken driver whose 972 kg sports car ran into a stationary 2094 kg station wagon stopped at a red traffic light. The cars stuck together and slid with locked wheels for 11.0 m before coming to rest. The coefficient of sliding friction on the dry road was 0.70. Estimate the speed of the sports car when it hit the station wagon.
Estimate the instantaneous acceleration of the sports car during the actual collision if the colliding cars reach the same final speed after collapsing a combined total distance of 0.8 m.
To estimate the speed of the sports car when it hit the station wagon, we can use the principle of conservation of momentum.
1. Calculate the initial momentum of the system:
The initial momentum of the system is the sum of the momentum of the sports car and the station wagon before the collision.
Initial momentum = (mass of sports car) * (initial velocity of sports car) + (mass of station wagon) * (initial velocity of station wagon)
Since the station wagon is stationary, the initial velocity of the station wagon is 0. The equation becomes:
Initial momentum = (mass of sports car) * (initial velocity of sports car)
2. Calculate the final momentum of the system:
The final momentum of the system is the sum of the momentum of the sports car and the station wagon after the collision.
Final momentum = (mass of sports car + mass of station wagon) * (final velocity)
Since the cars stuck together and moved together after the collision, the final velocity is the same for both cars.
3. Apply the law of conservation of momentum:
According to the principle of conservation of momentum, the initial momentum and the final momentum of a system should be equal if no external forces are acting.
Initial momentum = Final momentum
(mass of sports car) * (initial velocity of sports car) = (mass of sports car + mass of station wagon) * (final velocity)
Now, plug in the given values:
(mass of sports car) = 972 kg
(mass of station wagon) = 2094 kg
(initial velocity of station wagon) = 0 m/s
Solving for the initial velocity of the sports car, we get:
972 kg * (initial velocity of sports car) = (972 kg + 2094 kg) * (final velocity)
(initial velocity of sports car) = (2094 kg * (final velocity)) / 972 kg
To determine the final velocity of the combined cars, we need to calculate the work done due to friction during the slide.
4. Calculate the work done due to friction:
The work done by friction is given by the equation:
Work = (force of friction) * (distance)
Since the cars slid with locked wheels, the friction force is given by:
Force of friction = (coefficient of sliding friction) * (normal force)
The normal force can be calculated as:
Normal force = (mass of sports car + mass of station wagon) * (acceleration due to gravity)
So,
Force of friction = (coefficient of sliding friction) * (normal force)
Work = (force of friction) * (distance)
Now, plug in the given values:
(coefficient of sliding friction) = 0.70
(distance) = 11.0 m
Calculate the normal force:
Normal force = (mass of sports car + mass of station wagon) * (acceleration due to gravity)
Normal force = (972 kg + 2094 kg) * (9.8 m/s^2)
Now, calculate the force of friction and the work done due to friction.
5. Calculate the final velocity:
To calculate the final velocity, we'll use the work-energy principle.
The work done by friction is equal to the change in kinetic energy of the system:
Work = Change in kinetic energy
Change in kinetic energy = (1/2) * (mass of sports car + mass of station wagon) * (final velocity^2 - initial velocity of sports car^2)
Now, plug in the given and calculated values:
Work = (force of friction) * (distance)
Change in kinetic energy = (1/2) * (mass of sports car + mass of station wagon) * (final velocity^2 - initial velocity of sports car^2)
Solve the equation for the final velocity:
(final velocity) = sqrt((2 * Work) / (mass of sports car + mass of station wagon) + (initial velocity of sports car)^2)
Plug in the given and calculated values from step 4 to calculate the final velocity.
To estimate the instantaneous acceleration of the sports car during the actual collision, we can use the following formula:
Acceleration = (change in velocity) / (time taken)
In this case, the colliding cars reach the same final speed after collapsing a combined total distance of 0.8 m. The total distance covered by the cars during the actual collision can be calculated as:
Total distance = (distance before collision) - (distance after collision)
Total distance = 11.0 m - 0.8 m
Now, we can calculate the time taken during the collision:
(time taken) = (total distance) / (final velocity)
Finally, calculate the instantaneous acceleration:
Acceleration = (final velocity - initial velocity) / (time taken)
Plug in the given and calculated values to determine the instantaneous acceleration during the actual collision.