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 mechanical energy of a system is the sum of its potential energy (PE) and kinetic energy (KE). According to the law of conservation of energy, the total mechanical energy remains constant as long as no external forces are acting on the system. In other words, the initial mechanical energy is equal to the final mechanical energy.

Mathematically, we can express this as:

PEi + KEi = PEf + KEf

Let's break down the problem step by step:

Step 1: Express the initial and final states in terms of potential energy and kinetic energy.

For the initial state:
PEi = mgh (where m is the mass of the roller coaster, g is the acceleration due to gravity, and h is the initial height)
KEi = 0.5mv^2 (where v is the initial velocity)

For the final state:
PEf = mgh' (where h' is the final height)
KEf = 0.5mv'^2 (where v' is the final velocity)

Step 2: Plug in the given values into the equations.

Given:
h = 40 m
v = 15 m/s
h' = 5 m

Step 3: Solve for the final velocity.

Using the conservation of energy equation, we have:

mgh + 0.5mv^2 = mgh' + 0.5mv'^2

Plugging in the values:

m(9.8 m/s^2)(40 m) + 0.5mv^2 = m(9.8 m/s^2)(5 m) + 0.5mv'^2

Next, we can eliminate the mass (m) from both sides of the equation. Since it appears on both sides, it cancels out.

(9.8 m/s^2)(40 m) + 0.5(15 m/s)^2 = (9.8 m/s^2)(5 m) + 0.5v'^2

(392 N) + 0.5(225 m^2/s^2) = (49 N) + 0.5v'^2

392 N + 112.5 m^2/s^2 = 49 N + 0.5v'^2

Step 4: Rearrange the equation and solve for v'.

0.5v'^2 = (392 N + 112.5 m^2/s^2) - (49 N)

0.5v'^2 = 455.5 N + 112.5 m^2/s^2 - 49 N

0.5v'^2 = 418.5 N + 112.5 m^2/s^2

v'^2 = (418.5 N + 112.5 m^2/s^2) / 0.5

v'^2 = 837 N + 225 m^2/s^2

Finally, take the square root of both sides to find the final velocity (v'):

v' = √(837 N + 225 m^2/s^2)

Therefore, the velocity of the roller coaster at a height of 5 m above the ground can be found by taking the square root of (837 N + 225 m^2/s^2).