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?

Question

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?

Answer 1

To solve this problem, we need to use Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In this case, the acceleration will be negative since the roller coaster is slowing down.

First, let's convert the weight and the resistance force into acceleration due to Newton's second law. The weight of the roller coaster is 3640 lb, which we can convert to mass using the formula: mass = weight/acceleration due to gravity. We'll assume the acceleration due to gravity is 32 ft/s^2. Thus:
mass = 3640 lb / 32 ft/s^2

Next, we'll subtract the air-resistance force and the frictional force from the net force acting on the roller coaster.
net force = weight - air-resistance force - frictional force

Now, we can use the net force to calculate the acceleration.
acceleration = net force / mass

Given that the roller coaster is coasting to a stop at its starting location, we know its final velocity will be 0. Using the kinematic equation v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity (which we need to find), a is the acceleration, and s is the distance traveled, we can solve for the initial velocity.

In this case, the distance traveled is 0.57 mile. To convert this to feet, we know that 1 mile is equal to 5280 ft. Thus: distance traveled = 0.57 mile x 5280 ft/mile

Rearranging the kinematic equation, we have:
u^2 = v^2 - 2as

Since the final velocity is 0, the equation simplifies to:
u^2 = -2as

Substituting the values we found earlier, we have:
u^2 = -2 (acceleration) (distance traveled)

We can now solve for the initial velocity (u).

So, to summarize, the steps to solve this problem are as follows:

1. Convert the weight of the roller coaster to mass.
2. Calculate the net force acting on the roller coaster.
3. Calculate the acceleration.
4. Convert the distance to feet.
5. Use the kinematic equation to solve for the initial velocity.

Please note that I will need the values for the air-resistance force and the frictional force to proceed with the calculations.