Given the displacement function $s(t) = 2\sin(\pi t) + 2\cos(\pi t)$, we need to calculate the average velocity over various time intervals and then estimate the instantaneous velocity at $t = 1$.

(a) Average Velocity Calculation

The average velocity over a time interval $[t_1, t_2]$ is given by: $\text{Average velocity} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}$ where $s(t)$ is the displacement function.

(i) For the interval $[1, 2]$:

$s(1) = 2\sin(\pi \times 1) + 2\cos(\pi \times 1) = 2\sin(\pi) + 2\cos(\pi) = 2(0) + 2(-1) = -2 \text{ cm}$ $s(2) = 2\sin(\pi \times 2) + 2\cos(\pi \times 2) = 2\sin(2\pi) + 2\cos(2\pi) = 2(0) + 2(1) = 2 \text{ cm}$ $\text{Average velocity} = \frac{2 - (-2)}{2 - 1} = \frac{4}{1} = 4 \text{ cm/s}$

(ii) For the interval $[1, 1.1]$:

$s(1.1) = 2\sin(\pi \times 1.1) + 2\cos(\pi \times 1.1) = 2\sin(1.1\pi) + 2\cos(1.1\pi)$ $s(1.1) = 2(-0.309) + 2(0.951) \approx -0.618 + 1.902 = 1.284 \text{ cm}$ $\text{Average velocity} = \frac{1.284 - (-2)}{1.1 - 1} = \frac{3.284}{0.1} = 32.84 \text{ cm/s}$

(iii) For the interval $[1, 1.01]$:

$s(1.01) = 2\sin(\pi \times 1.01) + 2\cos(\pi \times 1.01) = 2\sin(1.01\pi) + 2\cos(1.01\pi)$ $s(1.01) = 2(-0.0314) + 2(0.9995) \approx -0.0628 + 1.999 = 1.9362 \text{ cm}$ $\text{Average velocity} = \frac{1.9362 - (-2)}{1.01 - 1} = \frac{3.9362}{0.01} = 393.62 \text{ cm/s}$

(iv) For the interval $[1, 1.001]$:

$s(1.001) = 2\sin(\pi \times 1.001) + 2\cos(\pi \times 1.001) = 2\sin(1.001\pi) + 2\cos(1.001\pi)$ $s(1.001) = 2(-0.00314) + 2(0.999995) \approx -0.00628 + 1.99999 = 1.99371 \text{ cm}$ $\text{Average velocity} = \frac{1.99371 - (-2)}{1.001 - 1} = \frac{3.99371}{0.001} = 3993.71 \text{ cm/s}$

(b) Instantaneous Velocity at $t = 1$

The instantaneous velocity at $t = 1$ can be estimated by finding the derivative of $s(t)$ with respect to $t$ and then evaluating it at $t = 1$.

First, find $s'(t)$: $s(t) = 2\sin(\pi t) + 2\cos(\pi t)$ $s'(t) = 2\pi\cos(\pi t) - 2\pi\sin(\pi t)$

At $t = 1$: $s'(1) = 2\pi\cos(\pi) - 2\pi\sin(\pi) = 2\pi(-1) - 2\pi(0) = -2\pi \text{ cm/s} \approx -6.28 \text{ cm/s}$

Final Answers:

(a) Average velocity:

• (i) [1, 2]: $4$ cm/s
• (ii) [1, 1.1]: $32.84$ cm/s
• (iii) [1, 1.01]: $393.62$ cm/s
• (iv) [1, 1.001]: $3993.71$ cm/s

(b) Instantaneous velocity at $t = 1$: $-6.28$ cm/s

Would you like more details or have any questions?

Related Questions:

1. How do you calculate average velocity for a different function?
2. What are the physical interpretations of positive and negative instantaneous velocities?
3. How would the results change if the displacement function included a quadratic term?
4. What happens to the velocity as the time interval decreases further?
5. How can you use limits to find the instantaneous velocity more rigorously?

Tip:

Always ensure to keep track of units when performing calculations, especially when dealing with physical quantities like displacement, velocity, and time.