Eric is standing on the ground, a distance of 70 ft from the bottom of Ferris wheel that has a 20 ft radius. His arm is at the same height as the bottom of the Ferris wheel. Janice is on Ferris wheel which makes one revolution counter clock wise every 16 secs. At the instant she is at a point A, Eric throws a ball to her at 60 ft/ sec at an angle of 60 degrees above the horizontal.will the ball reach Jenice? Show the equations for the Ferris wheel and the ball and draw the graph.
What is the minimum distance between the ball and Janiceon the Ferris wheel and what is the time at which the minimum distance occurs?
If Janice is at (0,0) when t=0, then the wheel obeys
w(t) = 20(1-cos(π/8 t))
The ball follows the parametric equations
x(t) = -70 + 60cos60° t
y(t) = 60sin60° t - 16t^2
I have no idea where point A is, but having the equations should make things amenable to solution.
Oops. The w(t) given above is the y-value of Janice's position. You also need to use the x-value, which I'm sure you can do.
I need the minimum distance
To determine if the ball will reach Janice on the Ferris wheel, we need to calculate the trajectory of the ball and analyze its position relative to Janice's position on the Ferris wheel at different times.
First, let's define the equations for the Ferris wheel and the ball:
Equation for the Ferris Wheel:
The Ferris wheel has a radius of 20 ft and completes one revolution counterclockwise every 16 seconds. We can express the height of Janice on the Ferris wheel as a function of time using the equation of vertical motion:
h(t) = 20 + 20 * sin((2π/16) * t)
The term 20 * sin((2π/16) * t) represents the vertical displacement of Janice as she moves around the Ferris wheel.
Equation for the Ball:
The ball is thrown with an initial velocity of 60 ft/sec at an angle of 60 degrees above the horizontal. We can analyze the x and y components of its motion separately.
For the x-component, the ball's horizontal motion is unaffected by gravity, so it travels at a constant speed:
x(t) = 60 * cos(60°) * t
For the y-component, we need to consider the effects of gravity. We use the equation of vertical motion:
y(t) = 0 + 60 * sin(60°) * t - (1/2) * 32.2 * t^2
Here, the term 60 * sin(60°) * t represents the initial vertical velocity, and the term (1/2) * 32.2 * t^2 represents the effect of gravity pulling the ball downwards.
Using these equations, we can plot the graph of Janice's position on the Ferris wheel and the trajectory of the ball. The x-axis represents time (t), and the y-axis represents the height (h) for the Ferris wheel, and the distance (y) for the ball.
To find the minimum distance between the ball and Janice on the Ferris wheel, we can calculate the distance between their positions as a function of time using the equation:
distance(t) = sqrt((x(t) - 70)^2 + (y(t) - h(t))^2)
We need to find the time at which the minimum distance occurs by finding the derivative of the distance function with respect to time, setting it equal to zero, and solving for time. The resulting time value will give us the time at which the minimum distance occurs.
After calculating the distance function and its derivative, you can draw the graph and determine the minimum distance and the associated time.