A pair of kids and a pair of adults decided to compete in a three-legged race. The kids got to start 33 meters ahead of the adults, since they had shorter legs. When they were told... A pair of kids and a pair of adults decided to compete in a three-legged race. The kids got to start 33 meters ahead of the adults, since they had shorter legs. When they were told to start, the kids hobbled forward at a rate of 1 meter per second, and the adults hobbled after them at a rate of 4 meters per second. Soon they were side-by-side. How far did the adults go? How long did that take?
Understand the Problem
The question describes a scenario involving a three-legged race between kids and adults. The kids start with a head start of 33 meters and move at 1 meter per second, while the adults move at 4 meters per second. The problem asks to find the distance covered by the adults until they are side-by-side with the kids, and how long that takes.
Answer
The adults went 44 meters, and it took them 11 seconds.
Answer for screen readers
The adults went 44 meters, and it took them 11 seconds.
Steps to Solve
- Define variables
Let $t$ be the time (in seconds) it takes for the adults to catch up with the kids. Let $d_a$ be the distance covered by the adults. Let $d_k$ be the distance covered by the kids.
- Express the distances in terms of time
The adults' distance is their speed multiplied by time: $d_a = 4t$ The kids' distance is their head start plus their speed multiplied by time: $d_k = 33 + 1t$
- Set the distances equal to each other
When the adults and kids are side-by-side, they have covered the same distance from the starting line of the adults. Therefore, the distance covered by the adults equals the initial head start of the kids plus the distance covered by the kids in time $t$.
$d_a = d_k$ $4t = 33 + t$
- Solve for $t$
Subtract $t$ from both sides: $3t = 33$ Divide by 3: $t = 11$ seconds.
- Calculate the distance the adults traveled
Substitute $t = 11$ into the equation for the adults' distance: $d_a = 4t = 4 * 11 = 44$ meters.
The adults went 44 meters, and it took them 11 seconds.
More Information
This is a classic problem of relative motion. An alternative approach would have been calculating the relative speed between the adults and the kids i.e. $4 - 1 = 3$ m/s and then computing the time required to close the initial distance of 33 meters at a relative speed of 3 m/s: $33 / 3 = 11$ seconds
Tips
A common mistake is forgetting to add the head start to the distance covered by the kids. Another mistake would be to calculate the distance of the kids instead of the adults.
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