Question

13 larry and marcus were jogging along a \\(100 \times 60\text{ m}\\) rectangular track. larry started from point a and marcus from point b. given that their speeds were \\(5\text{ m/s}\\) and \\(4\text{ m/s}\\) respectively, when did larry finally catch up with marcus?

Explanation:

Identify track dimensions and initial positions

The track is a rectangle with dimensions \(100\text{ m} \times 60\text{ m}\).
Point A and Point B are adjacent corners along the longer side of length \(100\text{ m}\).
Larry starts at Point A, moving clockwise (indicated by the arrow pointing right).
Marcus starts at Point B, moving clockwise (indicated by the arrow pointing left, which continues the clockwise path along the top edge).
The initial distance Larry must cover to catch up to Marcus along the clockwise path is the length of the side AB, which is \(100\text{ m}\).

Determine speeds and relative speed

Larry's speed is \(v_L = 5\text{ m/s}\).
Marcus's speed is \(v_M = 4\text{ m/s}\).
Since both are running clockwise in the same direction, this is a pursuit problem.
The relative speed at which Larry gains on Marcus is:

$$ v_{\text{rel}} = v_L - v_M = 5\text{ m/s} - 4\text{ m/s} = 1\text{ m/s} $$

Calculate the time required to catch up

Larry needs to close the initial gap of \(d = 100\text{ m}\) between Point A and Point B.
Using the relative speed formula:

$$ t = \frac{d}{v_{\text{rel}}} = \frac{100\text{ m}}{1\text{ m/s}} = 100\text{ s} $$

Verify the position of catch-up

In \(100\text{ s}\), Larry runs:

$$ d_L = 5\text{ m/s} \times 100\text{ s} = 500\text{ m} $$

The perimeter of the rectangular track is:

$$ P = 2 \times (100\text{ m} + 60\text{ m}) = 320\text{ m} $$

Larry's position after \(500\text{ m}\) from Point A:

$$ 500\text{ m} = 1 \times 320\text{ m} + 180\text{ m} $$

Starting from A and going clockwise:

• \(100\text{ m}\) reaches B.
• Another \(60\text{ m}\) (total \(160\text{ m}\)) reaches the third corner.
• Another \(20\text{ m}\) (total \(180\text{ m}\)) puts him on the bottom side of length \(100\text{ m}\).

Since this is a continuous loop, Larry successfully catches up with Marcus after \(100\text{ seconds}\).

Answer:

\(100\text{ seconds}\)