Question

Determine the required height h of the crest of the roller coaster to thebottom so that when it is essentially at rest at the crest of the hill it will reacha speed of 28 m/s when it comes to the bottom.

Answer 1

h = 40 m

Step-by-step explanation:

• Assuming no friction present, total mechanical energy must be conserved, so the following expression stands:
• ΔK + ΔU = 0 (1)
• Now, if the car is at rest at the crest of the hill, the change in kinetic energy is just as follows:

`\Delta K = (1)/(2) * m* v_(b) ^(2)` (2)

where vb = speed at the bottom = 28 m/s

• If we define the bottom as our zero reference level for the gravitational potential energy, we can write the following equation:

`\Delta U = U_(f) - U_(i) = 0- m*g*h = -m*g*h` (3)

• From (1) we get:
• ΔK = -ΔU
• Replacing by (2) and (3), we get:

`(1)/(2) * m* v^(2) = m*g*h`

• Simplifying and rearranging terms, we can solve for h (height required) as follows:

`h = (v_(b) ^(2) )/(2*g) = ((28m/s)^(2))/(2*9.8m/s2) = 40 m`