A 315 kg roller coaster starts from rest at the top of a 30-m hill. Assuming the track is frictionless, what is roller coaster's speed at the bottom of the hill?
To solve this problem, we can use the principle of conservation of mechanical energy. At the top of the hill, the roller coaster has gravitational potential energy, and at the bottom of the hill, it has kinetic energy.
The potential energy (PE) at the top of the hill is given by the formula:
PE = mgh
Where:
m = mass of the roller coaster = 315 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height of the hill = 30 m
PE = 315 kg * 9.8 m/s^2 * 30 m
PE = 91,980 Joules
According to the conservation of mechanical energy, this potential energy is converted into kinetic energy at the bottom of the hill.
The kinetic energy (KE) at the bottom of the hill is given by the formula:
KE = 1/2 mv^2
Where:
m = mass of the roller coaster = 315 kg
v = velocity of the roller coaster at the bottom of the hill (what we need to find)
Setting the potential energy equal to the kinetic energy:
PE = KE
91,980 Joules = 1/2 * 315 kg * v^2
Rearranging the equation to solve for v:
v^2 = 2 * 91,980 Joules / 315 kg
v^2 = 2 * 291.61904761904765 m^2/s^2
v^2 = 583.2380952380952 m^2/s^2
v ≈ √(583.2380952380952) ≈ 24.16 m/s
Therefore, the speed of the roller coaster at the bottom of the hill is approximately 24.16 m/s.
To find the roller coaster's speed at the bottom of the hill, we can use the principle of conservation of mechanical energy, which states that the total mechanical energy of a system remains constant as long as only conservative forces are acting on it.
The total mechanical energy of the roller coaster at the top of the hill is equal to the sum of its potential energy and kinetic energy:
E_top = PE_top + KE_top
Where PE_top is the potential energy and KE_top is the kinetic energy at the top of the hill.
The potential energy of an object of mass m at height h is given by the formula:
PE = m * g * h
Where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
In this case, the mass of the roller coaster is 315 kg, and the height of the hill is 30 m, so the potential energy at the top of the hill is:
PE_top = 315 kg * 9.8 m/s^2 * 30 m = 911,700 J
Since the roller coaster starts from rest at the top of the hill, its initial kinetic energy is zero:
KE_top = 0 J
Therefore, the total mechanical energy at the top of the hill is:
E_top = PE_top + KE_top = 911,700 J + 0 J = 911,700 J
According to the conservation of mechanical energy, the total mechanical energy at the top of the hill is equal to the total mechanical energy at the bottom of the hill:
E_top = E_bottom
In this case, at the bottom of the hill, all the potential energy is converted into kinetic energy:
E_bottom = PE_bottom + KE_bottom = 0 J + KE_bottom
Therefore, we can write:
0 J + KE_bottom = 911,700 J
Solving for KE_bottom, we find:
KE_bottom = 911,700 J
The kinetic energy is given by the formula:
KE = (1/2) * m * v^2
Where m is the mass of the roller coaster and v is its velocity.
Substituting the known values into the equation, we can solve for v:
911,700 J = (1/2) * 315 kg * v^2
Dividing both sides of the equation by (1/2) * 315 kg, we get:
v^2 = (2 * 911,700 J) / (315 kg)
v^2 = 5794.29 m^2/s^2
Taking the square root of both sides of the equation, we find:
v = √5794.29 m^2/s^2
v ≈ 76 m/s
Therefore, the roller coaster's speed at the bottom of the hill is approximately 76 m/s.
To find the roller coaster's speed at the bottom of the hill, we can use the principle of conservation of energy. At the top of the hill, the roller coaster has gravitational potential energy due to its elevation, which is converted into kinetic energy at the bottom of the hill.
The potential energy (PE) of an object at a certain height h is given by the formula: PE = m * g * h, where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
Therefore, at the top of the hill, the potential energy of the roller coaster is:
PE = 315 kg * 9.8 m/s^2 * 30 m = 92,610 J
At the bottom of the hill, this potential energy is completely converted into kinetic energy (KE).
The kinetic energy (KE) of an object is given by the formula: KE = 1/2 * m * v^2, where v is the velocity of the object.
Setting the potential energy equal to the kinetic energy:
PE = KE
92,610 J = 1/2 * 315 kg * v^2
Simplifying the equation:
92,610 J = 157.5 kg * v^2
Dividing both sides of the equation by 157.5 kg:
v^2 = 92,610 J / 157.5 kg
v^2 ≈ 588 m^2/s^2
Taking the square root of both sides of the equation:
v ≈ √(588 m^2/s^2)
v ≈ 24.2 m/s
Therefore, the roller coaster's speed at the bottom of the hill is approximately 24.2 m/s.