"A roller coaster reaches the top of the steepest hill with a speed of 1.4m/s. It then descends the hill which is at an average angle of 45 degrees and is 50m long. What will its speed be when it reaches the bottom?"

Assume conservation of mechanical energy (which means neglecting friction). The elevation change of the roller coaster is 50 sin 45 = 35.36 m

The Potential Energy loss equals the Kinetic Energy gain.

Oh thank you so much!

To find the speed of the roller coaster when it reaches the bottom of the hill, we can use the principle of conservation of mechanical energy.

1. First, let's calculate the potential energy (PE) at the top of the hill using the height and the mass of the roller coaster. Since the speed is given, there is no need to calculate the kinetic energy (KE) at the top of the hill.

2. Next, we will calculate the potential energy (PE') at the bottom of the hill using the height and the mass of the roller coaster.

3. By setting the PE at the top equal to the PE' at the bottom (neglecting any energy losses due to friction or air resistance), we can solve for the velocity of the roller coaster at the bottom of the hill.

Now let's break it down and calculate each step:

1. Calculate the potential energy (PE) at the top of the hill:
PE = m * g * h

Where:
m = mass of the roller coaster (given or assumed)
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height from the top of the hill to the bottom (height = length of the hill * sin(angle))

Here, we can calculate the height using the given angle and length:
height = 50 m * sin(45°) ≈ 35.4 m

Now, substitute the known values into the equation to find the potential energy at the top of the hill:
PE = m * g * h
= m * 9.8 m/s^2 * 35.4 m

2. Calculate the potential energy (PE') at the bottom of the hill:
Since the roller coaster is at the bottom of the hill, its height is 0, and the potential energy is 0.

PE' = 0

3. Set the PE at the top equal to PE' at the bottom and solve for the velocity (v') at the bottom:
PE = PE'
m * g * h = 0.5 * m * v'^2

Cancel out the mass (m) on both sides of the equation, and solve for the velocity (v'):
g * h = 0.5 * v'^2
v'^2 = 2 * g * h
v' = √(2 * g * h)

Substitute the known values into the equation:
v' = √(2 * 9.8 m/s^2 * 35.4 m)

Calculating the final value, we get:
v' ≈ 24.2 m/s

Therefore, the speed of the roller coaster when it reaches the bottom of the hill is approximately 24.2 m/s.