Answer:
A) X(t) = 2.78t + 0.24cos 2πt
& Y(t) = 0.24 - 0.24sin2πt
B) 1000(n-0.5) ≤ r ≤ 1000n
Foe values of n at 1, and 2, the speed is between 500 & 1000 and between 1500 & 2000 respectively. For higher values of n, we can calculate the range of r from the formula.
Step-by-step explanation:
To find the parametric equation, we need to find the vector for horizontal and vertical movements. Let's call it (x(t), y(t))
Since there is no vertical movement of bicycle, the initial component is;
(0, 0.24) (since it's centred 34cm above ground)
Now let's convert 10km/h to m/s;
Hence, 10km/h = 2.78 m/s
So vector of this in horizontal direction is; (2.78t,0)
Now, the circular motion component for vector with radius 0.24m is; (0.24cos 2πt, - 0.24Sin 2πt).
Now, when we add the 3 components to get;
X(t) = 0 + 2.78t + 0.24cos 2πt
= 2.78t + 0.24cos 2πt
And Y(t) = 0.24 + 0 - 0.24sin2πt
= 0.24 - 0.24sin2πt
For motion to appear backwards, we should only see half a revolution in his eye period. Therefore, for this, n ≥ 1, eye period is (n-0.5)P
Where P is the light period.
From research, human eye can see up to 1000 frames per second; therefore, converting it to seconds per frame, we have ; 1/1000 = 0.001
Hence,
(n-0.5)P ≤ 0.001 ≤ nP
Multiplying each term by 1000/P to get;
1000(n-0.5) ≤ 1/P ≤ 1000n
1/P is called rate of revolution (r)
So, 1000(n-0.5) ≤ r ≤ 1000n
So that's the formula when n ≥ 1.
When n is 1, r is between 500 and 100. When n is 2, r is between 1500 and 2000. And so it goes for values of n greater than 1.
