4.一列火车通过440米的桥需要40秒,以同样的速度穿过310米的隧道需要30秒.这列火车的速度和车身长各是多少?

To solve this problem, we need to find the speed of the train and its length using the given data. Here’s a step-by-step breakdown:

### Given:

1. **Bridge:**
   - Length of bridge = 440 meters
   - Time to pass the bridge = 40 seconds

2. **Tunnel:**
   - Length of tunnel = 310 meters
   - Time to pass the tunnel = 30 seconds

### Objective:

Find the speed of the train and the length of the train.

### Solution:

#### Definitions:

- \( v \) = speed of the train in meters per second (m/s)
- \( L \) = length of the train in meters

#### Calculations:

1. **Train passing the bridge:**

   - When the train passes the bridge, it covers the length of the bridge plus its own length in 40 seconds.
   - Distance covered = Length of bridge + Length of train = \( 440 + L \)
   - Time = 40 seconds

   Using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \), we have:
   \[
   440 + L = v \times 40
   \]

   Simplifying:
   \[
   v = \frac{440 + L}{40}
   \]

2. **Train passing the tunnel:**

   - When the train passes the tunnel, it covers the length of the tunnel plus its own length in 30 seconds.
   - Distance covered = Length of tunnel + Length of train = \( 310 + L \)
   - Time = 30 seconds

   Using the same formula:
   \[
   310 + L = v \times 30
   \]

   Simplifying:
   \[
   v = \frac{310 + L}{30}
   \]

#### Setting the equations equal:

Since both expressions are equal to \( v \), set them equal to each other:

\[
\frac{440 + L}{40} = \frac{310 + L}{30}
\]

Cross-multiply to solve for \( L \):

\[
30(440 + L) = 40(310 + L)
\]

\[
13200 + 30L = 12400 + 40L
\]

Rearrange and solve for \( L \):

\[
13200 - 12400 = 40L - 30L
\]

\[
800 = 10L
\]

\[
L = 80 \text{ meters}
\]

#### Find the speed \( v \):

Substitute \( L = 80 \) back into the speed equation:

Using \( v = \frac{440 + L}{40} \):

\[
v = \frac{440 + 80}{40} = \frac{520}{40} = 13 \text{ m/s}
\]

### Conclusion:

- The length of the train is \( 80 \) meters.
- The speed of the train is \( 13 \) meters per second (m/s).

If you have any further questions or need additional explanations, feel free to ask!

Here are some related questions:

1. What is the relationship between speed, distance, and time?
2. How do we convert speed from meters per second to kilometers per hour?
3. What other types of problems can be solved using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \)?
4. How do we solve problems involving multiple objects moving at different speeds?
5. What are some common mistakes when solving distance-speed-time problems?
6. How can we calculate average speed for a journey with different segments?
7. What is the effect of acceleration on the speed of an object?
8. How does relative speed work in the context of two objects moving towards or away from each other?

**Tip:** When solving problems involving multiple equations, it's often helpful to set expressions equal to each other to find a common variable.

Learn how to calculate the speed and length of a train using distance, speed, and time relationships with a step-by-step solution. This problem involves solving equations with variables and is suitable for high school level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation