The position function of the object is given as:

$s(t) = 45t^2 - 109t + 22$

To determine where the object is speeding up and slowing down, we need to analyze the velocity and acceleration of the object. Here's the process:

Step 1: Find the velocity function

The velocity is the first derivative of the position function with respect to time $t$:

$v(t) = \frac{ds(t)}{dt} = \frac{d}{dt}(45t^2 - 109t + 22)$

$v(t) = 90t - 109$

Step 2: Find the acceleration function

The acceleration is the derivative of the velocity function:

$a(t) = \frac{dv(t)}{dt} = \frac{d}{dt}(90t - 109)$

$a(t) = 90$

Step 3: Analyze speeding up and slowing down

• The object is speeding up when the velocity and acceleration have the same sign (either both positive or both negative).
• The object is slowing down when the velocity and acceleration have opposite signs.

Since the acceleration $a(t) = 90$ is constant and positive, we only need to analyze the velocity function $v(t) = 90t - 109$.

1. Set the velocity equal to zero to find when the object changes direction:

$v(t) = 0 \quad \Rightarrow \quad 90t - 109 = 0$ $t = \frac{109}{90} \approx 1.211$

Step 4: Interpret the results

• For $t < 1.211$, $v(t) < 0$ (the object is moving in the negative direction).
• For $t > 1.211$, $v(t) > 0$ (the object is moving in the positive direction).

Since the acceleration is always positive, the object is:

• Slowing down when $t < 1.211$ (because $v(t)$ is negative and $a(t)$ is positive).
• Speeding up when $t > 1.211$ (because $v(t)$ is positive and $a(t)$ is positive).

Conclusion:

• The object is slowing down in the interval $[0, 1.211]$.
• The object is speeding up in the interval $[1.211, 5]$.

Would you like more details on any part of this process?

Here are five related questions for further exploration:

1. How would the analysis change if the acceleration was not constant?
2. What is the object's velocity at $t = 0$ and $t = 5$?
3. How far has the object traveled by $t = 5$?
4. What would happen if the position function had a cubic term?
5. How would you find the maximum or minimum position of the object?

Tip: To better understand motion, always analyze both velocity and acceleration functions. Velocity tells you the direction and speed of motion, while acceleration reveals changes in that speed.