TRIG 11 SNOOPY AND CHARLIE
Snoopy and Charlie are going to go on two different Ferris wheels. One wheel is smaller than the other. Snoopy accesses the smaller wheel by a ramp which is 2 m off the ground. The wheel reaches a maximum height of 15 m. Charlie accesses a larger Ferris wheel by a ramp that is 5.5 m above ground and reaches a maximum height of 29 m above ground. The two wheels are separated by the ghost ship ride. When in range Snoopy is capable of springing onto Charlie's wheel by jumping across the ghost ship. At other times on the ride he cannot. Both rides last 2 minutes.
They both got on their Ferris wheels at the lowest point and at the same time. Snoopy's Ferris wheel is going around 2 revolutions/minute and Charlie's, 3 revolutions in 2 minutes. Sketch a diagram showing the wave forms for both graphs verified with your calculator. Find the times at which Snoopy is within range of Charlie. Charlie can catch Snoopy if he is ascending (rising) and is more than 8 m off the ground but less than 18 m off the ground.
AMPLITUDE
Snoopy: (max - min) /2 = (15 - 2)/2 = 6.5 m
Charlie: (max - min) /2 = (29 - 5.5)/2 = 11.75 m
Charlie: (max - min) /2 = (29 - 5.5)/2 = 11.75 m
FREQUENCY
The frequency B is equal to the distance of 2π divided by the time for one period.
Snoopy's wheel , is: 2π/30 = π/15 so B = π/15
Charlie's wheel : is 2π/40 = π/20 so B = π/20
Snoopy's wheel , is: 2π/30 = π/15 so B = π/15
Charlie's wheel : is 2π/40 = π/20 so B = π/20
HORIZONTAL SHIFT
Shift = - (time to maximum - 1/4 of period)
The negative cosine wave that we have been sketching to describe the motion of the wheel is the same as a sine wave that has been translated to the right on the x axis. It has moved by 1/4 or π/2.
For Snoopy, this means that distance along the x axis happens to be 1/4 x one period = 1/4 x 30 = 7.5 seconds. So the value of C = 7.5 and is written as (t - 7.5)
For Charlie, this means that distance along the x axis happens to be 1/4 x one period = 1/4 x 40 = 10 seconds. So the value of C = 10 and is written as (t - 10)
The negative cosine wave that we have been sketching to describe the motion of the wheel is the same as a sine wave that has been translated to the right on the x axis. It has moved by 1/4 or π/2.
For Snoopy, this means that distance along the x axis happens to be 1/4 x one period = 1/4 x 30 = 7.5 seconds. So the value of C = 7.5 and is written as (t - 7.5)
For Charlie, this means that distance along the x axis happens to be 1/4 x one period = 1/4 x 40 = 10 seconds. So the value of C = 10 and is written as (t - 10)
VERTICAL SHIFT
Both Ferris wheels are at different heights,
Snoopy's wheel = (max + min )/2 = (15 + 2)/2 = 8.5 m above the ground, so D = 8.5
Charlie's = (max + min )/2 = (29 + 5.5)/2 = 17.25 m above the ground, so D = 17.25
Snoopy's wheel = (max + min )/2 = (15 + 2)/2 = 8.5 m above the ground, so D = 8.5
Charlie's = (max + min )/2 = (29 + 5.5)/2 = 17.25 m above the ground, so D = 17.25
THE EQUATIONS
Snoopy: h(t) = 6.5sin(π/15(t - 7.5)) + 8.5
Charlie: h(t) = 11.75sin(π/20(t - 10)) + 17.25
Charlie: h(t) = 11.75sin(π/20(t - 10)) + 17.25