Question

The 800 -lb roller-coaster car starts from rest on the track having the shape of a cylindrical helix. If the helix descends 8ft for every one revolution, determine the time required for the car to attain a speed of 60ft/s. Neglect friction and the size of the car.

Answer 1

The question involves using conservation of energy principles to calculate the time a roller-coaster car takes to reach a speed of 60ft/s on a frictionless helical track. Potential energy converted to kinetic energy is used to find the height fallen, which, combined with the vertical descent per revolution, gives the number of revolutions and thus the time.

Step-by-step explanation:

The student is asking about the motion of a roller-coaster car down a helical track, using principles from Physics related to motion, specifically under the influence of gravity without friction.

We are asked to calculate the time it takes for the car to reach a certain speed (60ft/s), assuming it starts from rest.

To solve this problem, we can use the principle of conservation of energy.

The potential energy (PE) lost by the car as it descends the helical track is converted into kinetic energy (KE) as it speeds up.

The potential energy lost is equal to the weight of the car (W = 800lb) multiplied by the vertical distance (h) it falls.

Since the car starts from rest, its initial kinetic energy is zero.

Using the relationship PE_{initial} + KE_{initial} = PE_{final} + KE_{final}, we get W × h = \frac{1}{2} × m × v^{2}.

The velocity (v) we're solving for is 60ft/s, and the mass (m) of the roller-coaster car needs to be converted from pounds to slugs using the conversion factor 1 slug = 32.2 lb️.

After calculating the height fallen using the speed and mass, we can find the time taken to reach this speed by dividing the height fallen by the vertical descent per revolution, giving us the number of revolutions. Using the relationship velocity = 2️ππ (where r is the radius of the helix and T is the period), we can solve for time.

Answer 2

The time required for the roller-coaster car to attain a speed of 60 ft/s can be determined using the equations of motion. The time required is approximately 30π seconds.

Step-by-step explanation:

The time required for the roller-coaster car to attain a speed of 60 ft/s can be determined using the equations of motion. The initial velocity is 0 ft/s since the car starts from rest. The final velocity is 60 ft/s. The acceleration can be calculated using the formula = 8 ft/rev.

Using the equation = + ×, where is the initial velocity, is the acceleration, is the time, and is the displacement, we can rearrange the equation to solve for. In this case, is the initial velocity, is the final velocity, is the acceleration, and is the displacement. Rearranging the equation gives us:

60 ft/s = 0 ft/s + (8 ft/rev) ×

Now we can solve for :

= (60 ft/s) / (8 ft/rev) = 7.5 rev/s

Next, we need to convert the revolutions per second to seconds. Since there are 2π radians in one revolution, we can multiply 7.5 rev/s by 2π rad/rev to get the angular velocity in radians per second:

angular velocity = (7.5 rev/s) × (2π rad/rev) = 15π rad/s

Finally, we can find the time required for the car to attain a speed of 60 ft/s by dividing the displacement (8 ft/rev) by the linear velocity (60 ft/s) and multiplying by the angular velocity (15π rad/s):

time = (8 ft/rev) × (15π rad/s) / (60 ft/s) = 30π s