A 3640 lb roller coaster experiences a constant air-resistance force of 26 lb and a constant friction force of 20 lb. What initial velocity is required for the coaster to make it around the 0.57 mile track and coast to a stop at its starting location?
The initial kinetic energy must equal the friction work done in traveling 0.57 miles (3010 feet).
If you want the speed in feet/second, the mass must be in slugs, and the distance X must be in feet.
3640 lb = 113 slugs
(M/2) V^2 = 46 lb*X
Solve for V.
V^2 = 2*46*3010/113 = 2451 ft^2/s^2
If you are not familiar with the use of slugs for mass, do the problem with force, mass and distance in SI units, to get the velocity in m/s.
Thank you very much
To solve this problem, we need to apply Newton's second law of motion which states that the net force acting on an object is equal to its mass times its acceleration. In this case, the net force is the sum of the forces acting on the roller coaster.
The net force acting on the roller coaster can be calculated by subtracting the air-resistance force and friction force from the weight of the roller coaster:
Net force = Weight - Air resistance - Friction
Given data:
Weight = 3640 lb
Air resistance = 26 lb
Friction = 20 lb
Now, let's calculate the net force:
Net force = 3640 lb - 26 lb - 20 lb
= 3594 lb
Since the roller coaster is supposed to coast to a stop at its starting location, its final velocity will be zero. The initial velocity required can be found using the kinematic equation:
v^2 = u^2 + 2as
where,
v = final velocity (in this case, 0)
u = initial velocity (unknown)
a = acceleration
s = distance
Given data:
v = 0
a = Net force / mass (from Newton's second law)
s = 0.57 mile (converted to feet)
Now, we can rearrange the equation to solve for u:
u = sqrt( -2as )
Substituting the known values:
u = sqrt( -2 * (3594 lb) * (0.57 mile * 5280 ft/mile) )
Calculating this expression will give you the required initial velocity for the roller coaster to make it around the track and coast to a stop at its starting location.