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Answer :
The distance the car coasts upward as a function of the number of people is approximately 3.06 meters.
To determine the distance the car coasts upward as a function of the number of people (n), we need to analyze the energy conservation in the system.
Initially, the elevator car is moving upward at a speed of 3.00 m/s, and we need to find the distance it coasts upward before coming to a stop.
The total initial mechanical energy of the system is given by the sum of the kinetic energy of the car, counterweight, and people in the car:
E_initial = (1/2)mv_car^2 + (1/2)mv_counterweight^2 + n(1/2)mv_person^2
where m is the mass of each person, v_car is the initial velocity of the car, v_counterweight is the initial velocity of the counterweight (which is assumed to be zero), and v_person is the velocity of each person in the car (which is assumed to be the same as the car's velocity).
At the final moment when the motor is disconnected, the car coasts freely without friction. It will come to a stop when its kinetic energy is completely converted into potential energy. The final mechanical energy of the system is therefore given by:
E_final = nmgd
where g is the acceleration due to gravity and d is the distance the car coasts upward.
Since energy is conserved, we can equate the initial and final energies:
E_initial = E_final
(1/2)mv_car^2 + (1/2)mv_counterweight^2 + n(1/2)mv_person^2 = nmgd
Simplifying and solving for d, we get:
d = [(v_car^2 + v_counterweight^2 + 2v_person^2) / (2g)]
Substituting the given values and assuming the counterweight initially at rest, we have:
d = [(3.00^2 + 0 + 2(3.00^2)) / (2(9.8))]
Simplifying further, we find:
d = 3.06 meters
Therefore, the distance the car coasts upward as a function of the number of people is approximately 3.06 meters.
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