A roller-coster car starts from rest at the top of a 16 meter hight track, and slides without friction through a 6 meter tall loop. Find the speed of the car at the bottom of the loop. find the speed of the car when it's upside down at the top of the loop
The answer depends upon the elevation of the 6 meter loop, relative to the starting elevation. That is not clear from your decription.
In any case, use conservation of energy.
By the way, not all cars in a roller coaster "train" are at the same altitude most of the time. It is the CM elevation of the entire train that matters
To find the speed of the car at the bottom of the loop, we can use the principle of conservation of mechanical energy. At the top of the track, the car has gravitational potential energy, which is converted into kinetic energy at the bottom of the loop.
First, let's find the potential energy at the top of the track. The potential energy is given by the equation:
Potential Energy = mass × acceleration due to gravity × height
Since the car is at rest, its initial kinetic energy is zero. Therefore, the initial mechanical energy is equal to the potential energy:
Initial Energy = Potential Energy = mass × acceleration due to gravity × height
Next, let's find the velocity of the car at the bottom of the loop using the equation for kinetic energy:
Kinetic Energy = (1/2) × mass × velocity²
At the bottom of the loop, all the potential energy is converted into kinetic energy, so we have:
Potential Energy = Kinetic Energy
Plugging in the values and solving for velocity:
mass × acceleration due to gravity × height = (1/2) × mass × velocity²
acceleration due to gravity × height = (1/2) × velocity²
2 × acceleration due to gravity × height = velocity²
velocity = √(2 × acceleration due to gravity × height)
Substituting the known values: acceleration due to gravity ≈ 9.8 m/s² and height = 16 m, we can calculate the speed at the bottom of the loop.
velocity = √(2 × 9.8 m/s² × 16 m)
velocity = √(313.6 m²/s²)
velocity ≈ 17.7 m/s
So, the speed of the car at the bottom of the loop is approximately 17.7 m/s.
To find the speed of the car when it's upside down at the top of the loop, we need to consider the forces acting on the car. The only force acting is the gravitational force, which is directed towards the center of the loop. This force provides the centripetal acceleration needed to keep the car moving in a circle.
The centripetal force is given by the equation:
Centripetal Force = mass × (velocity)² / radius
At the top of the loop, the centripetal force is provided solely by gravity, so we have:
mass × (velocity at the top of the loop)² / radius = mass × acceleration due to gravity
Simplifying the equation:
(velocity at the top of the loop)² = radius × acceleration due to gravity
Plugging in the known values: radius = 6 m and acceleration due to gravity ≈ 9.8 m/s², we can calculate the speed at the top of the loop.
(velocity at the top of the loop)² = 6 m × 9.8 m/s²
(velocity at the top of the loop)² = 58.8 m²/s²
velocity at the top of the loop ≈ √58.8 m/s
velocity at the top of the loop ≈ 7.7 m/s
So, the speed of the car when it's upside down at the top of the loop is approximately 7.7 m/s.