We can solve this problem by setting up a system of equations based on the relationship between speed, time, and distance.

Step 1: Define variables

• Let the speed of the plane in still air be $p$ km/h.
• Let the speed of the wind be $w$ km/h.

Step 2: Use the information given

On the flight to Kampala (with a tailwind):

• The plane’s speed relative to the ground is the sum of the speed of the plane in still air and the speed of the wind: $\text{Speed to Kampala} = p + w = 158 \text{ km/h}.$

On the return trip (with a headwind):

• The plane’s speed relative to the ground is the difference between the speed of the plane in still air and the speed of the wind: $\text{Speed back} = p - w = 112 \text{ km/h}.$

Step 3: Set up the system of equations

We now have the following system of two equations: $p + w = 158 \quad \text{(1)}$ $p - w = 112 \quad \text{(2)}.$

Step 4: Solve the system of equations

Add equations (1) and (2) to eliminate $w$: $(p + w) + (p - w) = 158 + 112,$ $2p = 270,$ $p = \frac{270}{2} = 135 \text{ km/h}.$

Now, substitute $p = 135$ into equation (1) to find $w$: $135 + w = 158,$ $w = 158 - 135 = 23 \text{ km/h}.$

Final Answer:

The speed of the wind is 23 km/h.

Would you like further clarification on any of the steps? Here are some additional questions related to this problem:

1. How would the result change if the plane had an even higher speed in still air?
2. What would happen if the wind speed were to increase significantly?
3. How can we apply this method to solve similar problems with different distances or speeds?
4. How would the time for each trip change if the wind speed were different?
5. What are some real-world scenarios where these types of calculations are applied?

Tip: To solve problems involving relative motion (like this one), always break down the motion in terms of the speed relative to the ground and the object’s speed in still air.