Question

1. A roller coaster car has a mass of 250 Kg. The car is towed to the top of a 30 m hill where it is released from rest and allowed to roll.   The car plunges down the hill, then up an 8 m hill and through a loop with a radius of 8 m. Assuming no friction: (8 points)

1. What is the potential energy of the car at the top of the 30 m hill?

2. What are the Kinetic Energy and the speed of the car at the bottom of the 30 m hill?

3. What are the Kinetic Energy and the speed of the car at the top of the 8 m hill?

4. If the hill makes an angle of 60 ° with the horizontal and the car takes 15 seconds to be towed up the 30 m hill, determine the length of the hill, the velocity of the car, the force required to tow the car up the hill, and the power of the motor pulling the car up the hill.

Please type your answer, Please do not write on the Paper and Post t.

Answer 1

Mass of the roller coaster car given = 250 kg

height of the top of the hill from ground (bottom ) = 30 m

A) potential energy of the car at the top of the 30 m hill = m*g*h

= 250 *9.8 *30

= 73500 J

B) from conservation of energy principle we have (K.E + P.E)_Bottom = (K.E +P.E) _Top

{since K.E_Top = 0 ( it is momentarily at rest) and P.E_Bottom = 0 }

∴ K.E_bottom = P.E_top

also total Potential energy is converted to kinetic energy as the car descends to the bottom of the hill .i.e from minimum .zero to maximum kinetic energy of 73500 J.

so at the bottom of the hill the kinetic energy = 73500 J

speed of the car at the bottom of the 30 m hill = ?

K.E = 1/2 *M*V²

73500 = 1/2 *M*V²

V² = 73500*2 /250

= 588

V = 24.25 m/s

C) we know P.E = K.E

change in potential energy results in the conversion of kinetic energy

so, m*g*(H) = 1/2*m*V²

V² = 2 **9.8 *(30-8)

= 431.2

so V =(431.2)^0.5

= 20.77 m/s   

∴ speed of the car at the top of the 8 m hill = 20.77 m/s

and magnitude of kinetic energy at the top of the 8 m hill =1/2 *m*V²

= 1/2*250*(20.77²⁾

= 53924.11 J

D) If the hill makes an angle of 60 ° with the horizontal  the length of the hill, = 30 /sin 60 °

= 34.64 m

velocity of the roller coaster car = length / time

= 34.64 /15

= 2.31 m/s

Force required to tow the car up , F = m*a

= m *( V-U) /t

= 250*( 2.31 -0) /15

= 38.5 N

power = force *velocity

= 38.5 *2.31

= 88.94 Watts