Matthew is riding a ferris wheel at a constant speed of 10km/h. The boarding height for the wheel is 1m, and the wheel has a radius of 7m. What is the equation for the function that describes Matthews height in the terms of time, assuming Matthew starts at the highest point on the wheel
center of wheel is 1 + 7 = 8 m above ground
cos (2 pi t/T) is max of +1 when t = 0
so h = 8 + 7 cos (2 pi t/T)
what is T, the period or time to go 2 pi radians?
speed = 10,000 meters/3600 seconds
= 2.78 m/s
distance around = 2 pi r = 2 pi*7 = 44 m
so
T = 44m/2.78m/s = 15.8 seconds
so
h = 8 + 7 cos (2 pi t/15.8)
Well, Matthew sure is going to have a "wheel" good time! Let's see if we can tackle this height equation, shall we?
Since Matthew starts at the highest point on the wheel, we know that his initial height is 1 meter. As the wheel rotates, let's say it takes t hours for Matthew to complete one revolution.
Now, we know that the circumference of a circle (or the path Matthew takes) can be calculated using the formula: C = 2πr. In this case, the radius (r) of the ferris wheel is 7 meters. So, the circumference is C = 2π * 7 = 14π meters.
Since Matthew is riding at a constant speed of 10 km/h, we need to convert that to meters per hour: 10 km/h = 10,000 meters / 1 hour.
Now, we want to find Matthew's height as a function of time. We can start by saying that, for every t hours that pass, he completes one full revolution. This means that his position on the wheel increases by the circumference of the wheel.
So, if we let h(t) represent Matthew's height at time t, we can write the equation as:
h(t) = 1 + (10,000 meters / 1 hour) * t / (14π meters)
Voila! That's the equation for Matthew's height in terms of time while riding the ferris wheel. Now, let's hope he enjoys the ride and doesn't get too dizzy up there!
To find an equation that describes Matthew's height in terms of time while riding the ferris wheel, we can start by considering the properties of circular motion.
The height of an object moving in a circular path is determined by the vertical component of its position. In this case, the vertical component is the height above the ground. The equation for the vertical position of an object moving in a circular path can be written as:
y(t) = R + r*cos(ωt),
where:
- y(t) is the vertical position (height) at time t,
- R is the initial height (1m in this case),
- r is the radius of the circular path (7m in this case),
- ω is the angular velocity (rate of change of angle with respect to time).
In this scenario, Matthew starts at the highest point on the wheel, which means he is at the maximum height position. Therefore, the equation becomes:
y(t) = 1 + 7*cos(ωt).
Now, we need to determine the value of ω. The angular velocity ω can be calculated using the formula:
ω = v / r,
where:
- v is the linear velocity (speed),
- r is the radius.
Given that Matthew is riding the ferris wheel at a constant speed of 10 km/h (which is equivalent to 10,000 m/60 minutes or 166.67 m/min), and the radius is 7m, we can substitute these values into the formula:
ω = 166.67 m/min / 7m.
ω ≈ 23.81 min⁻¹.
Now, we can substitute the value of ω back into the equation:
y(t) = 1 + 7*cos(23.81t).
Therefore, the equation for the function that describes Matthew's height in terms of time is y(t) = 1 + 7*cos(23.81t).
To derive the equation for Matthew's height as a function of time, we need to understand the relationship between height, time, and the motion of the ferris wheel.
First, let's establish a few things:
- Let "h" represent Matthew's height above the ground at any given time "t".
- At the highest point on the ferris wheel, Matthew's height is at a maximum of 1m.
- The ferris wheel has a radius of 7m, which means that its vertical motion follows a circular path with a radius of 7m.
Since the ferris wheel is moving at a constant speed, we can relate Matthew's height "h" to the angular displacement "θ" of the ferris wheel.
Let's assume that when Matthew starts, the ferris wheel is at the highest point on its path. We can then determine the relationship between "h" and "θ" as follows:
- When θ = 0° (starting point), h = 1m.
- When θ = 180° (halfway point), h = 8m (1m from the highest point plus 7m from the radius of the ferris wheel).
- When θ = 360° (complete revolution), h = 1m (back to the highest point).
From this, we can deduce that the height "h" is given by the equation:
h = 7 * sin(θ) + 1
However, we still need to determine θ in terms of time "t". Since the ferris wheel completes one full revolution (360°) in a certain amount of time, we can relate "θ" to "t" using the concept of angular velocity.
The angular velocity "ω" is defined as the angle covered per unit time. Since the ferris wheel moves at a constant speed of 10km/h, and the circumference of a circle is given by 2πr, we have:
10km/h = (2π * 7m) * ω
Simplifying this equation, we find:
ω = (5π/7) rad/h
Now, integrating the angular velocity over time will give us the angular displacement "θ" as a function of time:
θ = ∫[(5π/7) rad/h] dt
θ = (5π/7)t
Substituting this expression for θ into the equation for "h", we get:
h = 7 * sin((5π/7)t) + 1
Therefore, the equation that describes Matthew's height "h" in terms of time "t" is:
h = 7 * sin((5π/7)t) + 1