Let’s solve the problems step-by-step.

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Problem 1

Find the values of $t$ for which the velocity is increasing in $s = t^3 - 6t^2 + 4$.

Solution:

1. Find the velocity, $v(t)$: $v(t) = \frac{ds}{dt} = 3t^2 - 12t$

2. Find the acceleration, $a(t)$: $a(t) = \frac{dv}{dt} = 6t - 12$

3. Velocity is increasing when $a(t) > 0$: $6t - 12 > 0 \quad \implies \quad t > 2$

   Answer: Velocity is increasing for $t > 2$.

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Problem 2

Find the velocity and acceleration when $t = 2$ if $s = \sqrt{8t} + \sqrt[3]{4t}$.

Solution:

1. Find the velocity, $v(t)$: $v(t) = \frac{ds}{dt} = \frac{1}{2\sqrt{8t}}(8) + \frac{1}{3\sqrt[3]{4t^2}}(4)$ Simplify: $v(t) = \frac{4}{\sqrt{8t}} + \frac{4}{3\sqrt[3]{4t^2}}$

   At $t = 2$: [ v(2) = \frac{4}{\sqrt{16}} + \frac{4}{3\sqrt[3]{16}} = 1 + \frac{4}{3\sqrt[3]{16}} ]

2. Find the acceleration, $a(t)$: Differentiate $v(t)$: $a(t) = -\frac{4}{2\sqrt{8t^3}}(8) + \frac{8}{9\sqrt[3]{4t^5}}$

   Substitute $t = 2$ to compute $a(2)$.

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Problem 3

$s = t^4 - 6t^3 + 12t^2 - 10t + 3$.

a) When is the speed increasing and decreasing?

1. Find $v(t)$: $v(t) = 4t^3 - 18t^2 + 24t - 10$

2. Find $a(t)$: $a(t) = 12t^2 - 36t + 24$

   Solve $a(t) > 0$ to find when speed is increasing.

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b) When does the direction of motion change?

1. Direction changes when $v(t) = 0$.

   Solve $4t^3 - 18t^2 + 24t - 10 = 0$ for $t$.

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c) Find the total distance traveled in the first 3 seconds:

1. Evaluate $s(0)$, $s(1)$, $s(2)$, and $s(3)$.

2. Compute $|s(1) - s(0)| + |s(2) - s(1)| + |s(3) - s(2)|$.

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Problem 4

The distance of a locomotive from a fixed point is $s = 3t^4 + 144t^2$. When is it in reverse?

Solution:

1. Find $v(t)$: $v(t) = \frac{ds}{dt} = 12t^3 + 288t$

2. The locomotive is in reverse when $v(t) < 0$.

   Factorize: $12t(t^2 + 24) < 0$

   Since $t^2 + 24 > 0$ always: $v(t) < 0 \quad \text{for} \quad t < 0$

   Answer: The locomotive is never in reverse for $t \geq 0$.

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Follow-up Questions:

1. How would the solutions change if additional forces were acting on the particle?
2. Can you find critical points of $s$ for each problem to determine when motion changes?
3. How can we extend this analysis to include jerk (rate of change of acceleration)?
4. How does total distance differ from displacement?
5. What are the real-world applications of rectilinear motion?

Tip: Always start by finding derivatives systematically and check conditions carefully.