Final Answer:
The speed of the car is approximately 16.21 m/s.
Step-by-step explanation:
To calculate the speed of the car, we can use the principle of conservation of energy. This principle states that the total mechanical energy of a closed system remains constant. In the case of the roller coaster car, the only forces acting on the car are gravity and the normal force of the track. Therefore, the total mechanical energy of the car is conserved.
We can break down the total mechanical energy of the car into two components: potential energy and kinetic energy. The potential energy of the car is due to its height above the ground, and the kinetic energy of the car is due to its motion. At the highest point of the track, the car has only potential energy. As the car descends the track, its potential energy is converted into kinetic energy.
Using the principle of conservation of energy, we can write the following equation:
PE_initial + KE_initial = PE_final + KE_final
where:
PE_initial is the potential energy of the car at the highest point of the track
KE_initial is the kinetic energy of the car at the highest point of the track (which is zero since the car is at rest)
PE_final is the potential energy of the car at the bottom of the track (which is zero since the car is at ground level)
KE_final is the kinetic energy of the car at the bottom of the track
We can solve this equation for KE_final, which is equal to the car's speed at the bottom of the track:
KE_final = PE_initial
The potential energy of the car at the highest point of the track is equal to its mass times the acceleration due to gravity times its height above the ground:
PE_initial = m * g * h
where:
m is the mass of the car
g is the acceleration due to gravity (9.81 m/s^2)
h is the height of the car above the ground
Substituting this equation into the equation for KE_final, we get:
KE_final = m * g * h
The kinetic energy of an object is equal to one-half of its mass times its velocity squared:
KE_final = 1/2 * m * v^2
where:
v is the velocity of the car
Substituting this equation into the equation for KE_final, we get:
1/2 * m * v^2 = m * g * h
Solving this equation for v, we get:
v = sqrt(2 * g * h)
Plugging in the values for g and h (assuming the car starts from a height of 100 meters), we get the speed of the car at the bottom of the track:
v = sqrt(2 * 9.81 m/s^2 * 100 m)
v ≈ 16.21 m/s
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Complete Question
In the figure of a frictionless roller coaster car, what is the speed of the car?
Figure is attached below.
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