The figure below shows the path of a cart moving at various positions (A to E) along a smooth roller coaster. The 20.0 kg cart was projected at a height of 30.0 m above ground with an initial speed of 5.0 m/s.
(a) Calculate the speed at point B of the roller coaster ride.
(b) Calculate the maximum height (position C) the cart can reach. State the assumption made.
(c) Point D is the top of the circular loop. By indicating the force(s) acting on the cart at that point, explain why no work is done on the cart even though the cart is still moving at that point.
(d) Passengers have to be secured with safety belt. Throughout the ride, different forces will be exerted by the safety belt on the passengers to keep them on their seats. It is absolutely necessary at two points of the ride that the safety belt be working properly. State which are the 2 points and explain the choice of your points.
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Answer:
(a) Using the conservation of energy,
initial energy = final energy
½ m u2 + mgh = ½ m v2
½ u2 + gh = ½ v2
½ (5.0)2 + (10)(30.0) = ½ v2
½ v2 = 12.5 + 300
v2 = 625
v = 25 m/s
(b) Using the conservation of energy,
initial energy = final energy
½ m v2 = mgh
h = ½ v2 / g
h = ½ (25)2 / 10
h = 31.3 m
Assumptions:
1) No energy is lost due to air resistance
2) Cart reaches C with no speed.
(c)
The blue force is the force due to gravity.
The red force is the force due to the contact force from the path.
No work is done because the direction of motion is perpendicular to the direction of the force.
(d) Point C and Point E.
Point C because cart is moving down when inertia will move the body forward; the safety belt will pull the passenger down along with the cart as it moves down.
Point E because it is when the cart is stopping due to brakes. Inertia will cause the body to continue to move forward when the brakes are applied on the cart; the safety belt will exert a force to stop the passenger from being thrown forward out of the cart.
