Understanding the Car Speed Problem

This problem involves two cars, x and y, traveling between two places, A and B, which are 140 km apart. We are given information about their meeting times under two different scenarios: when they move in the same direction and when they move towards each other. We are also told that car y is faster than car x. Our goal is to find the speed of car y.

Setting Up the Equations

Let $S_x$ be the speed of car x in km/h and $S_y$ be the speed of car y in km/h. The distance between A and B is 140 km.

Scenario 1: Cars Move in the Same Direction

When the cars move in the same direction, they meet after 7 hours. Since car y starts from B and is faster than car x (which starts from A), for them to meet, car y must be moving away from A in the same direction as car x, and the meeting point will be beyond B. The relative speed when moving in the same direction is the difference between their speeds, $S_y - S_x$ (since $S_y > S_x$). The distance covered by this relative speed to meet is the initial distance between them, 140 km.

Using the formula: Distance = Speed $\times$ Time

Relative Distance = Relative Speed $\times$ Time

$$140 = (S_y - S_x) \times 7$$

Dividing both sides by 7:

$$S_y - S_x = \frac{140}{7}$$

$$S_y - S_x = 20 \quad (*)$$

Scenario 2: Cars Move Towards Each Other

When the cars move towards each other, they start from A and B and meet after 1 hour. The relative speed when moving towards each other is the sum of their speeds, $S_x + S_y$. The distance they cover together to meet is the initial distance between A and B, which is 140 km.

Using the formula: Distance = Speed $\times$ Time

Relative Distance = Relative Speed $\times$ Time

$$140 = (S_x + S_y) \times 1$$

$$S_x + S_y = 140 \quad (**)$$

Solving for the Speeds

We now have a system of two linear equations with two variables:

Equation (*): $S_y - S_x = 20$

Equation (**): $S_x + S_y = 140$

To find $S_y$, we can add the two equations together:

$$(S_y - S_x) + (S_x + S_y) = 20 + 140$$

$$S_y - S_x + S_x + S_y = 160$$

$$2S_y = 160$$

Now, divide by 2 to find $S_y$:

$$S_y = \frac{160}{2}$$

$$S_y = 80$$

The speed of car y is 80 km/h.

We can also find the speed of car x by substituting the value of $S_y$ into Equation (**):

$$S_x + 80 = 140$$

$$S_x = 140 - 80$$

$$S_x = 60$$

The speed of car x is 60 km/h.

Verification

Let's check if these speeds satisfy the conditions given in the problem:

• Is $S_y > S_x$? Yes, 80 km/h is greater than 60 km/h.
• If they move in the same direction (relative speed = $80 - 60 = 20$ km/h), do they meet after 7 hours? Time = Distance / Speed = $140 / 20 = 7$ hours. Yes.
• If they move towards each other (relative speed = $60 + 80 = 140$ km/h), do they meet after 1 hour? Time = Distance / Speed = $140 / 140 = 1$ hour. Yes.

All conditions are met.

Conclusion

The speed of car y is 80 km/h.

Revision Table: Distance, Speed, Time Concepts

Additional Information: Relative Motion Problems

Problems involving cars, trains, or people moving are common applications of distance, speed, and time. Understanding the concept of relative speed is crucial for solving these problems efficiently. When objects are moving relative to each other, their speeds are combined in different ways depending on their direction of movement.

• When objects move in the same direction, their relative speed is the difference between their individual speeds. This relative speed determines how quickly the distance between them changes. If the faster object is initially behind the slower one, this speed is how quickly it catches up. If the faster object is initially ahead, this speed is how quickly it pulls away. In our problem, car y (faster) started ahead of car x (from B relative to A), and they met as y moved further away from A, increasing the gap over the initial 140km until x reached the meeting point.
• When objects move in opposite directions (towards each other or away from each other), their relative speed is the sum of their individual speeds. When moving towards each other, this sum is the rate at which the distance between them decreases. When moving away from each other, this sum is the rate at which the distance between them increases.

By identifying the relative speed and the relevant distance (often the initial distance between the objects), you can use the standard Distance = Speed $\times$ Time formula to set up equations and solve for unknown speeds or times.