Distance Calculation for Buses: Speed and Time

This problem requires us to find the distance covered by two buses. We know their speeds and the difference in the time they take to cover the same distance.

Key Concepts: Speed, Distance, and Time

The relationship between distance ($d$), speed ($v$), and time ($t$) is fundamental in physics. The formula is:

$$d = v \times t$$

We can rearrange this formula to find time:

$$t = \frac{d}{v}$$

In this problem, two buses cover the same distance ($d$). Let's define the variables:

• Speed of the slower bus ($v_1$) = 40 km/h
• Speed of the faster bus ($v_2$) = 50 km/h
• Time taken by the slower bus = $t_1$
• Time taken by the faster bus = $t_2$

We are given that the slower bus takes one hour more than the faster bus. This translates to the equation:

$$t_1 = t_2 + 1$$

Setting Up the Equations

Since the distance ($d$) is the same for both buses, we can write:

1. For the slower bus: $d = v_1 \times t_1 = 40 \times t_1$
2. For the faster bus: $d = v_2 \times t_2 = 50 \times t_2$

Because both expressions equal $d$, we can set them equal to each other:

$$40 \times t_1 = 50 \times t_2$$

Solving for Time

Now we can substitute the time relationship ($t_1 = t_2 + 1$) into the equation:

$$40 \times (t_2 + 1) = 50 \times t_2$$

Distribute the 40 on the left side:

$$40 t_2 + 40 = 50 t_2$$

To solve for $t_2$, we can subtract $40 t_2$ from both sides:

$$40 = 50 t_2 - 40 t_2$$

$$40 = 10 t_2$$

Now, divide by 10:

$$t_2 = \frac{40}{10}$$

$$t_2 = 4 \text{ hours}$$

This means the faster bus takes 4 hours to cover the distance.

We can find the time taken by the slower bus using $t_1 = t_2 + 1$:

$$t_1 = 4 + 1 = 5 \text{ hours}$$

The slower bus takes 5 hours.

Calculating the Distance

Now that we have the times for both buses, we can calculate the distance ($d$) using the formula $d = v \times t$. We can use the values from either the slower or the faster bus.

Using the slower bus:

$$d = 40 \text{ km/h} \times 5 \text{ h} = 200 \text{ km}$$

Using the faster bus:

$$d = 50 \text{ km/h} \times 4 \text{ h} = 200 \text{ km}$$

Both calculations yield the same result, confirming the distance is 200 km.

Conclusion

The distance covered by each bus is 200 km. The slower bus (40 km/h) takes 5 hours, and the faster bus (50 km/h) takes 4 hours. The difference in time is indeed 1 hour.