A particle moves in the xy-plane in such a way that its position at time t is Bold r (t) = (t - Bold sin t) i + (1 - Bold cos t) j. Find the maximum and minimum values of |v| and |a|.

- The maximum value of |v| is 2

- The maximum value of |a| is 1

Step-by-step explanation:

Given r (t) = (t + sin t) i + (1 - cos t) j

We are required to find the maximum and minimum values of the magnitude of velocity, |v|, and magnitude of acceleration, |a|.

To do these, we need to first obtain the velocity and acceptation from the given function.

- To obtain velocity, differentiate r (t) with respect to t.

Doing that, we have

v (t) = (1 - cos t) i - sin t j

Note: If A = xi + yj + zk, Magnitude of A, denoted as |A| = √ (x² + y² + z²).

Knowing this,

|v| = √[ (1 - cos t) ² + sin²t]

= √[ (1 - 2cos t + cos²t) + sin²t]

= √ (2 - 2cost) (because a trig. identity, sin²t + cos²t = 1)

Now, the maximum value of |v| will be the maximum value of √ (2 - 2cost), which is when cos t = - 1, because 0 and 1 assume smaller values.

So, for cos t = - 1

√ (2 - 2cost) = √ (2 - 2 (-1))

= √4

max |v| = 2

- To obtain acceleration, a, we need to differentiate r twice. This is the same as differentiating v once.

a = dv/dt = sin t i - cos t j

|a| = √[ (sin t) ² + (-cos t) ²]

= √ (sin²t + cos²t)

= √1

max |a| = 1