You have a summer job working for a company that designs amusement park rides. T

Question

You have a summer job working for a company that designs amusement park rides. Their latest rollercoaster design involves a circular loop-the-loop with a radius of 9 meters. Neglecting friction, what's the minimum energy required for ensuring that the 1060 kg rollercoaster car is able to complete its trip around the loop?

One potential design for the rollercoaster involves having the car roll down from the top of a straight ramp (angled at 30 degrees above the horizontal) before it goes into the loop-the-loop. What is the minimum height required for this hill if there is a small friction coefficient of 0.10 for the full length of the hill?

An alternative design involves using a power supply to accelerate the car along a straight stretch of track. The power supply can deliver energy at a rate of 35 kW. How long should the straight track be assuming that it can be made approximately frictionless and that the car starts from rest?

Explanation / Answer

Here Power, P is constant

We know that ,

a = acceleration

s = distance/ displacement

Power is the rate of change of work with respect to time and the change in work is the change in kinetic energy

So power,P = d(KE)/dt = d(1/2*m*v2)/dt

= 1/2*m*d(v2)/dt

= 1/2*m*2*v*dv/dt.

But, dv/dt=ds/dt*dv/ds and ds/dt=v,

so dv/dt=v*dv/ds

making P=m*v2*dv/ds.

That makes P*ds=m*v2*dv.

Since power is constant, the integral of the left side is just P*s.

The integral of the right side is 1/3*m*v3.

So s=m*v3/(3*P) <-----------answer

Related Questions

Navigate

• Browse (All)
• Browse Y
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.