A and B are participants in a 1,800 m circular race. A is running at a speed of 24 km/h and B is running at a speed of 15 km/h. If both start
from the same point and at the same time and run in the same direction, find the time when they will meet again. (up to two decimal points)

Question

Prepp Team · Accepted Answer

Circular Race Meeting Time Calculation

This problem asks us to find the time it takes for two participants, A and B, to meet again at the starting point while running a 1,800 m circular race in the same direction. We are given their speeds and need to calculate the meeting time.

Problem Setup Analysis

Let's break down the information given:

Race Track Length: 1,800 meters (m)
Participant A's Speed ($v_A$): 24 kilometers per hour (km/h)
Participant B's Speed ($v_B$): 15 kilometers per hour (km/h)
Direction of Running: Same
Starting Conditions: Both start from the same point at the same time.

Calculating Speeds in Meters per Second

To solve this problem effectively, it's best to work with consistent units. Since the track length is in meters and we need the time in seconds, we should convert the speeds from km/h to meters per second (m/s).

The conversion formula is: $1 \text{ km/h} = \frac{1000 \text{ m}}{3600 \text{ s}} = \frac{5}{18} \text{ m/s}$.

Speed of A ($v_A$) in m/s: $$v_A = 24 \text{ km/h} \times \frac{5}{18} \text{ m/s per km/h}$$ $$v_A = \frac{24 \times 5}{18} \text{ m/s} = \frac{120}{18} \text{ m/s}$$ Simplifying the fraction by dividing numerator and denominator by 6: $$v_A = \frac{20}{3} \text{ m/s}$$
Speed of B ($v_B$) in m/s: $$v_B = 15 \text{ km/h} \times \frac{5}{18} \text{ m/s per km/h}$$ $$v_B = \frac{15 \times 5}{18} \text{ m/s} = \frac{75}{18} \text{ m/s}$$ Simplifying the fraction by dividing numerator and denominator by 3: $$v_B = \frac{25}{6} \text{ m/s}$$

Determining Relative Speed

When two objects move in the same direction, their relative speed is the difference between their individual speeds. This tells us how quickly the faster object is catching up to the slower one.

Participant A is faster than participant B. The relative speed ($v_{rel}$) at which A gains on B is:

$$v_{rel} = v_A - v_B$$ $$v_{rel} = \frac{20}{3} \text{ m/s} - \frac{25}{6} \text{ m/s}$$

To find the difference, we need a common denominator, which is 6:

$$v_{rel} = \left(\frac{20}{3} \times \frac{2}{2}\right) \text{ m/s} - \frac{25}{6} \text{ m/s}$$ $$v_{rel} = \frac{40}{6} \text{ m/s} - \frac{25}{6} \text{ m/s}$$ $$v_{rel} = \frac{40 - 25}{6} \text{ m/s}$$ $$v_{rel} = \frac{15}{6} \text{ m/s}$$ Simplifying the fraction by dividing numerator and denominator by 3: $$v_{rel} = \frac{5}{2} \text{ m/s} \text{ or } 2.5 \text{ m/s}$$

This means runner A gains 2.5 meters on runner B every second.

Time Calculation for Meeting

For A and B to meet again at the starting point, runner A must have completed exactly one full lap more than runner B. The distance A needs to gain on B is equal to the length of the race track.

Distance A needs to gain: 1,800 m
Relative Speed ($v_{rel}$): 2.5 m/s

We can find the time ($T$) using the formula: Time = Distance / Speed.

Time ($T$) when they meet again:

$$T = \frac{\text{Distance to gain}}{v_{rel}}$$ $$T = \frac{1800 \text{ m}}{2.5 \text{ m/s}}$$

Calculating the value:

$$T = \frac{1800}{5/2} \text{ seconds}$$ $$T = 1800 \times \frac{2}{5} \text{ seconds}$$ $$T = \frac{3600}{5} \text{ seconds}$$ $$T = 720 \text{ seconds}$$