One swimmer in a relay race has a 0.50 s lead and is...

One swimmer in a relay race has a 0.50 s lead and is swimming at a constant speed of $4.00 \mathrm{m} / \mathrm{s}$. The swimmer has $20.0 \mathrm{m}$ to swim before reaching the end of the pool. A second swimmer moves in the same direction as the leader. What constant speed must the second swimmer have in order to catch up to the leader at the end of the pool?

Key Concepts

Time Lag

Time lag refers to the delay between the start times of two events or moving objects. In many motion problems, one object may start its motion later than another. This difference in starting times must be taken into account when setting up the problem, as it affects how long each object has to travel a certain distance and consequently influences the calculation for when or where they might meet.

Uniform Motion

Uniform motion refers to movement at a constant speed in a straight line, meaning there is no acceleration or deceleration. In problems that involve uniform motion, the relationship between distance, speed, and time is linear, which allows the use of the formula distance = speed × time without the need to account for changing speeds.

Relative Motion

Relative motion involves analyzing the movement of one object with respect to another. In scenarios where two objects start at different times or positions, comparing their distances traveled over the same time period helps determine conditions for one object to overtake or catch up to the other.

Time, Speed, and Distance Relationship

The fundamental relationship between time, speed, and distance is central to solving kinematics problems. This relationship, expressed as distance = speed × time, allows one to set up equations for each moving object. By equating the time and distance for two objects, one can determine unknown variables such as the required speed needed for one object to catch up with another.

Transcript

00:01 In this problem we're going to talk about uniform motion.

00:03 Basically, all we need to remember is that for uniform motion, the velocity of an object is equal to the displacement divided by the time it took for the object to be displaced.

00:24 So basically what we have in our problem is a swimmer that is swimming with a speed of four meters per second.

00:33 Still has 20 meters left in his swimming race and he is 0 .5 seconds ahead of the next swimmer.

00:48 So he's in front of the second swimmer meaning that the second swimmer is located at a distance delta x prime of delta t times his speed v2 from the first swimmer.

01:09 So this is what being a distance of 0 .5 seconds from the second swimmer means.

01:15 It means that if the swimmer were to stop the first one, then the second swimmer would need 0 .5 seconds in order to reach the first swimmer.

01:28 This means that the total distance necessary still left for the second swimmer is equal to delta x prime, plus delta 1, delta x1.

01:40 Okay, so this is delta t v2 plus delta x1.

01:46 Also, the total time, t, that the first swimmer is going to take to finish his race is equal to delta x1 divided by v.

02:00 So that's 20 meters divided by 4 meters per second.

02:04 So this is five seconds.

02:07 Okay.

02:09 And what we have to calculate is what must.

02:11 Be the speed of the second swimmer such that he catches up with the first swimmer at the end of the race so basically we want them both to reach the final the finish line of the race at the same time and so basically what we have what we want actually is that delta x2 be equal to v2 times t okay and t here is five seconds...

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