How Do Acceleration and Speed Affect Race Outcome in Sprinting?

[Latios1314]

In a women's 100-m race, accelerating uniformly, Laura takes 2.00 s and Healan 3.00 s to attain their maximum speeds, which they each maintain for the rest of the race. They cross the finish line simultaneously, both setting a world record of 10.4 s.
(a) What is the acceleration of each sprinter?
(b) what are their respective maximum speeds?
(c) which sprinter is ahead at the 6.00-s mark, and by how much?
(d) What is the maximum distance by which Healan is behind Laura, and at what time does that occur?

I've managed to solve part a, b and c of the question but I'm stuck at part d.

I've difficulty coming up with the equations for the distance ran by Laura and Healan. I've stuck at trying to fit both the final velocities and the accleration of the runners into the same equation.

 

[davo789]

Hi Latios1314,

I haven't yet worked through the math, but I was wondering whether you're familiar with differentials?

 

[Latios1314]

I'm rather familiar with it but I'm hacing difficulty coming up with the distance formula's for both girls.

For the first three parts, i actually broke up the speed to two components, the constant velocity component and the acceleration phase.

I'm not really sure how I'm supposed to combine the two to form a distance formula wrt to time for part d.

 

[BruceW]

Yes, it was good to split it up into acceleration phase and constant velocity phase.

Think of it like a graph of velocity against time. For the constant acceleration phase, it will be a line with gradient equal to the acceleration. And for the constant speed phase, it will be a horizontal line.

So what is the distance at a particular time? just the total area under the graph up until that time!

Also, keep in mind that the graphs for each of the two people will look different, since they have different accelerations for different amounts of times, but I'm guessing you know that already since you did a,b,c

 

[Latios1314]

But how do I combine the two parts together into one equation?

 

[BruceW]

You're asking how to put together the acceleration phase and constant velocity phase to get one equation?

Well the final speed, position and time of the acceleration phase will equal the initial speed, position and time of the velocity phase. You need to make use of this to get the answer to part d.
But there isn't one single equation for the motion. There is one equation for during the acceleration phase and one equation for the constant velocity phase.