The WindSeeker ride at Canada's Wonderland lifts riders up to 300 feet into the air while spinning them around in a 12.2 m radius circle. The WindSeeker starts its circular motion from rest with a constant angular acceleration of 0.0174 rad/s2. It maintains that angular acceleration for 48.1 s, after which the angular speed remains constant for the rest of the ride - another 120. s. What is the top speed of the ride?
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Well, if we want to calculate the top speed of the ride, we need to take into account the different phases of its motion. First, the ride starts from rest and undergoes an angular acceleration for 48.1 seconds. After that, it maintains a constant angular speed for another 120 seconds.
Now, to find the top speed, we can break down the problem into two parts. Let's tackle the first part, where the ride is undergoing angular acceleration.
To find the final angular velocity (ω) during this period, we can use the formula:
ω = ω₀ + αt
Where:
ω = final angular velocity
ω₀ = initial angular velocity (which is 0 since the ride starts from rest)
α = angular acceleration (0.0174 rad/s²)
t = time (48.1s)
Plugging in the values:
ω = 0 + (0.0174 rad/s²)(48.1s)
Now calculate that and we'll continue.
To find the top speed of the ride, we need to calculate the angular speed first.
Given:
Initial angular velocity, ω₀ = 0 rad/s (starting from rest)
Angular acceleration, α = 0.0174 rad/s²
Time during acceleration, t₁ = 48.1 s
Time during constant speed, t₂ = 120 s
Using the kinematic equation for rotational motion:
ω = ω₀ + α * t
For the first part of the ride, during acceleration:
ω₁ = 0 + 0.0174 * 48.1
= 0.83694 rad/s
For the second part of the ride, during constant speed:
ω₂ = ω₁
= 0.83694 rad/s
Now, to find the top speed (linear speed), we can use the formula:
v = ω * r
where r is the radius of the circular motion.
Given:
Radius, r = 12.2 m
For the first and second parts of the ride:
v = ω * r₁ = ω₁ * r = 0.83694 * 12.2 = 10.209 m/s
v = ω * r₂ = ω₂ * r = 0.83694 * 12.2 = 10.209 m/s
Therefore, the top speed of the ride, during both the acceleration and constant speed, is 10.209 m/s.
To find the top speed of the WindSeeker ride, we need to first calculate the angular speed it reaches after the initial acceleration phase. Then we can convert that into linear speed using the given radius of the circular motion.
1. Calculate the angular speed attained during the acceleration phase:
Angular acceleration (α) = 0.0174 rad/s²
Time (t) = 48.1 s
Using the equation of motion for angular acceleration:
ω = ω0 + αt
Where:
ω = Final angular speed
ω0 = Initial angular speed (which is zero as the ride starts from rest)
Plugging in the values:
ω = 0 + (0.0174 rad/s²) * (48.1 s) = 0.83754 rad/s
2. Calculate the linear speed using the given radius:
Radius (r) = 12.2 m
Linear speed (v) = ω * r
Plugging in the values:
v = (0.83754 rad/s) * (12.2 m) = 10.2226 m/s
Therefore, the top speed of the WindSeeker ride is approximately 10.2226 m/s.