https://r2.erweima.ai/mathbot/image/17271948780122jkirog8.jpg

This problem describes two runners, A and B, who start from the same location. Runner A starts 30 seconds earlier than runner B, running at a constant speed of 5 m/s, while runner B runs at a constant speed of 6 m/s.

We are tasked to find:
a) The distance at which runner B will catch up to runner A.
b) The time it will take for runner B to catch up to runner A.

Let's define the problem in terms of equations.

### Step 1: Define the positions of the runners over time

1. **Runner A's position** after \( t \) seconds:
   \[
   x_A = 5(t + 30)
   \]
   This is because runner A has been running for \( t + 30 \) seconds by the time we start measuring.

2. **Runner B's position** after \( t \) seconds:
   \[
   x_B = 6t
   \]

### Step 2: Set the positions equal to find when they meet

Since runner B catches up to runner A when their positions are equal:
\[
5(t + 30) = 6t
\]

Now, let's solve for \( t \).The time it will take for runner B to catch up to runner A is 150 seconds.

### Step 3: Calculate the distance

Now that we know the time \( t = 150 \) seconds, we can substitute it into the equation for runner B's position to find the distance where the catch-up occurs:
\[
x_B = 6 \times 150 = 900 \, \text{meters}
\]

Thus, runner B will catch up to runner A after 150 seconds at a distance of 900 meters from the starting point.

### Final Answers:
a) The distance at which runner B catches up to runner A is **900 meters**.
b) The time it takes for runner B to catch up is **150 seconds**.

Would you like more details or have any questions?

Here are 5 related questions:
1. What if runner B had a lower speed than runner A, would they still meet?
2. How would the result change if both runners started at the same time but with different speeds?
3. What if runner A changed speed after a certain time?
4. How would this problem change if the runners followed a curved path?


Dos corredores A y B parten del mismo lugar. El competidor A parte 30 s antes que el corredor B, con una velocidad constante de 5 m/s. El participante B sigue la misma trayectoria del participante A con una velocidad constante de 6 m/s. ¿A qué distancia del punto de partida el corredor B alcanzará al corredor A? ¿En qué tiempo se realiza este alcance?

Learn how to calculate when and where one runner catches up to another when starting at different times and speeds, using linear motion and relative velocity equations. Suitable for high school students studying basic physics or applied mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation