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

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).