The problem involves the position of a car given by the equation $s(t) = 5t - 5t^2$, and it has three parts:

1. Find the displacement of the car between $t = 3 \, \text{s}$ and $t = 5 \, \text{s}$.
2. Determine the average velocity during the same time interval.
3. Find the instantaneous velocity at $t = 3 \, \text{s}$ and $t = 5 \, \text{s}$.

Let's solve each part step by step:

Solution

1. Displacement of the car

The displacement is the change in position of the car between $t_1 = 3$ and $t_2 = 5$.
The position function is:
$s(t) = 5t - 5t^2$

Evaluate $s(t)$ at $t = 3$ and $t = 5$:
$s(3) = 5(3) - 5(3^2) = 15 - 45 = -30 \, \text{m}$
$s(5) = 5(5) - 5(5^2) = 25 - 125 = -100 \, \text{m}$

The displacement is:
$\Delta s = s(5) - s(3) = -100 - (-30) = -100 + 30 = -70 \, \text{m}$

So, the displacement is $-70 \, \text{m}$.

2. Average velocity

The average velocity is given by:
$v_{\text{avg}} = \frac{\Delta s}{\Delta t}$
Here, $\Delta t = t_2 - t_1 = 5 - 3 = 2 \, \text{s}$, and $\Delta s = -70 \, \text{m}$.

$v_{\text{avg}} = \frac{-70}{2} = -35 \, \text{m/s}$

The average velocity is $-35 \, \text{m/s}$.

3. Instantaneous velocity

The instantaneous velocity is the derivative of $s(t)$:
$v(t) = \frac{d}{dt} \left( 5t - 5t^2 \right) = 5 - 10t$

Now evaluate $v(t)$ at $t = 3$ and $t = 5$:
$v(3) = 5 - 10(3) = 5 - 30 = -25 \, \text{m/s}$
$v(5) = 5 - 10(5) = 5 - 50 = -45 \, \text{m/s}$

The instantaneous velocities are:

• At $t = 3$: $-25 \, \text{m/s}$
• At $t = 5$: $-45 \, \text{m/s}$

Final Answers

1. Displacement: $-70 \, \text{m}$
2. Average velocity: $-35 \, \text{m/s}$
3. Instantaneous velocity:
   • At $t = 3$: $-25 \, \text{m/s}$
   • At $t = 5$: $-45 \, \text{m/s}$

Would you like detailed steps for any part or further clarifications?

5 Related Questions

1. How is displacement different from total distance traveled?
2. Can the average velocity ever equal the instantaneous velocity in any scenario?
3. What does a negative velocity indicate in the context of motion?
4. How would the calculations change if the equation included a constant term?
5. What is the significance of the slope of the $s(t)$ graph in interpreting velocity?

Tip

Always check the units of your answers to ensure consistency and correctness in physics problems.