A roller coaster is initially at a height of 40 m above the ground and has an initial velocity of 15 m/s. Using conservation of energy, find the velocity of the roller coaster at a height of 5 m above the ground.
To find the velocity of the roller coaster at a height of 5 m above the ground using conservation of energy, we need to consider the conservation of mechanical energy.
The conservation of mechanical energy states that the sum of the potential energy and the kinetic energy of an object remains constant, as long as no external forces are acting on the object.
At the highest point of the roller coaster (initial position), the roller coaster has potential energy (PE) due to its height above the ground and kinetic energy (KE) due to its motion. Let's denote the potential energy as PE1, the kinetic energy as KE1, and the total mechanical energy as E1:
E1 = PE1 + KE1
At a height of 5 m above the ground (final position), let's denote the potential energy as PE2, the kinetic energy as KE2, and the total mechanical energy as E2:
E2 = PE2 + KE2
Since energy is conserved, the total mechanical energy at the initial position (E1) is equal to the total mechanical energy at the final position (E2):
E1 = E2
Now, let's find the expressions for the potential and kinetic energies at each position. The potential energy (PE) of an object 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/s²), and h is the height of the object above a reference level.
At the initial position (height = 40 m), the potential energy is:
PE1 = m * g * h1 = m * 9.8 * 40
At the final position (height = 5 m), the potential energy is:
PE2 = m * g * h2 = m * 9.8 * 5
Now, let's consider the kinetic energy (KE) of the roller coaster. The kinetic energy of an object is given by the formula KE = 1/2 * m * v^2, where m is the mass of the object and v is its velocity.
At the initial position, the kinetic energy is given as KE1 = 1/2 * m * v1^2, where v1 = 15 m/s.
At the final position (where we want to find the velocity), the kinetic energy is given as KE2 = 1/2 * m * v2^2, where v2 is the unknown velocity.
Now we can set up the equations to solve for v2. Since the total mechanical energy is conserved, we have:
E1 = PE1 + KE1 = E2 = PE2 + KE2
Plugging in the values we've calculated:
(m * 9.8 * 40) + (1/2 * m * (15^2)) = (m * 9.8 * 5) + (1/2 * m * v2^2)
Now, we can solve this equation to find v2, the velocity of the roller coaster at a height of 5 m above the ground.