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Solve these two and give me logic behind them

Let's go through the solutions for both of these problems one by one, along with the logic behind each.

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### Problem 22:
**A car slows down from 23 m/s to rest in a distance of 85 m. What was its acceleration, assumed constant?**

#### Solution:
We are given:
- Initial velocity, \( v_i = 23 \, \text{m/s} \)
- Final velocity, \( v_f = 0 \, \text{m/s} \) (since the car comes to rest)
- Distance, \( d = 85 \, \text{m} \)

We need to find the acceleration, \( a \).

To solve this, we can use the following kinematic equation that relates velocity, distance, and acceleration:
\[
v_f^2 = v_i^2 + 2a d
\]
Where:
- \( v_f \) is the final velocity,
- \( v_i \) is the initial velocity,
- \( a \) is the acceleration, and
- \( d \) is the distance.

Plugging in the known values:
\[
0^2 = 23^2 + 2a \cdot 85
\]
\[
0 = 529 + 170a
\]
Now, solve for \( a \):
\[
170a = -529
\]
\[
a = \frac{-529}{170}
\]
\[
a \approx -3.11 \, \text{m/s}^2
\]

Thus, the car's acceleration is approximately \( -3.11 \, \text{m/s}^2 \). The negative sign indicates that the acceleration is in the opposite direction to the motion, meaning it's deceleration.

---

### Problem 23:
**A light plane must reach a speed of 33 m/s for takeoff. How long a runway is needed if the (constant) acceleration is 3.0 m/s²?**

#### Solution:
We are given:
- Final velocity, \( v_f = 33 \, \text{m/s} \)
- Initial velocity, \( v_i = 0 \, \text{m/s} \) (since the plane starts from rest)
- Acceleration, \( a = 3.0 \, \text{m/s}^2 \)

We need to find the distance (runway length), \( d \).

Using the same kinematic equation as in the previous problem:
\[
v_f^2 = v_i^2 + 2a d
\]
Substitute the known values:
\[
33^2 = 0^2 + 2 \cdot 3.0 \cdot d
\]
\[
1089 = 6d
\]
Now, solve for \( d \):
\[
d = \frac{1089}{6}
\]
\[
d = 181.5 \, \text{m}
\]

Thus, the plane needs a runway length of approximately **181.5 meters**.

---

### Key Concepts and Logic:
1. **Kinematic Equations**: In both problems, we used the kinematic equation that connects initial velocity, final velocity, acceleration, and distance. This is useful when time is not involved in the problem:
   \[
   v_f^2 = v_i^2 + 2ad
   \]
2. **Deceleration vs Acceleration**: In Problem 22, we see a negative acceleration because the car is slowing down, while in Problem 23, the acceleration is positive since the plane is speeding up.
3. **Solving for Variables**: In each case, the goal was to isolate the unknown variable (either acceleration or distance) and solve for it by substituting the known values into the equation.

---

Do you have any questions about these solutions or need further clarification?

Here are 5 related questions for further practice:
1. If a car slows down from 30 m/s to 0 in 100 m, what is its acceleration?
2. A train accelerates from rest at 2 m/s² for 20 seconds. What distance does it cover?
3. A plane takes off at a speed of 40 m/s and requires 200 m of runway. What is its acceleration?
4. How much time does it take for a car to slow from 25 m/s to 0 if it decelerates at \( 4 \, \text{m/s}^2 \)?
5. A cyclist speeds up from 5 m/s to 20 m/s with an acceleration of 2 m/s². What distance is covered during this acceleration?

**Tip**: Always double-check your units when solving physics problems to ensure consistency (e.g., m/s for speed, m/s² for acceleration).

Solve the following problems: 1. A car slows down from 23 m/s to rest in a distance of 85 m. What was its acceleration, assumed constant? 2. A light plane must reach a speed of 33 m/s for takeoff. How long a runway is needed if the constant acceleration is 3.0 m/s²?

Learn how to solve two kinematic problems: a car decelerating from 23 m/s over 85 m and calculating the runway length needed for a plane to reach 33 m/s with constant acceleration of 3.0 m/s². Includes step-by-step explanations of key concepts like acceleration and kinematic equations.

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