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

Answer:

h = 40 m

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:

       $\Delta K = \frac{1}{2} * m* v_{b} ^{2}$  (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:

       $\frac{1}{2} * m* v^{2} = m*g*h$

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

       $h = \frac{v_{b} ^{2} }{2*g} = \frac{(28m/s)^{2}}{2*9.8m/s2} = 40 m$