Step-by-step explanation:
To find the time required for the roller-coaster car to attain a speed of 60ft/s, we can use the kinematic equation for an object undergoing constant acceleration.
The cylindrical helix can be modeled as a path with circular motion components, and the car starts from rest. We'll calculate the time using the following steps:
Step 1: Find the radius of the helix
The distance the car travels for one revolution is the circumference of the helix. Given that the helix descends 8ft for every one revolution, the circumference of the helix can be calculated as 2 π × r, where r is the radius.
2 × π × r = 8
r = 8 / (2 × π) = 4 / π = 1.2732 ft (approx)
Step 2: Find the angular speed
The angular speed (ω) is the rate of change of the car's angular displacement. Since the car travels at a constant speed in a circular path, the relationship between linear speed (v) and angular speed (ω) is v = ω × r. Given the linear speed v and the radius r, we can find the angular speed using the formula:
60 ft/s = ω × 1.2732 ft
ω ≈ 47.1703 rad/s
Step 3: Find the time required
The relationship between linear speed, angular speed, and acceleration is v = ω × r = r dθ / dt, where θ is the angle, and t is the time. The centripetal acceleration "a" can be expressed as a = r ω^2.
At any given time, the speed of the car will be v = r × ω. We want to solve for the time required to attain a speed of 60ft/s. This can be done by integrating the acceleration over time, which gives:
v = r × ω = a × t
60 = 1.2732 × 47.1703 = 60t
t ≈ 1 second
Therefore, the time required for the car to attain a speed of 60ft/s is approximately 1 second.
