# Which runner has a larger maximum speed?

1. Feb 9, 2013

### lorenamtimas

1. The problem statement, all variables and given/known data

Suppose that two runners run a 100-meter dash, but the first runner reaches maximum speed more quickly than the second runner. Both runners maintain constant speed once they have reached their maximum speed and cross the finish line at the same time. Which runner has the larger maximum speed? Explain.

2. Relevant equations

No relevant equations.

3. The attempt at a solution
I understand that the second runner is the one who reaches a larger maximum speed. I just don't know how it relates to a velocity-time graph. The books explains that "if both runners cover the same distance in the same time interval, then their average velocity has to be the same and the area under the curves on a velocity-time graph are the same". Being that Conceptual Physics is the first physics class I have ever taken I can't grasp this concept. Please help me understand.

2. Feb 9, 2013

### DarkLightA

Alright.

On a velocity time graph, the area under the graph is distance traveled, since you multiply the two axes together.

So, as they both reach the finish at the same time, and the area under the graph (distance) is constant, the only factor that is changing is the velocity from point to point.

Now, in the beginning runner one accelerates quickly up to max speed and stays there. Runner two accelerates more slowly, so imagine his line on the velocity time graph being less steep. However, we determined that the area must be constant. How do we weight up for the fact that runner two has less area under the graph (distance covered) in the start? Well, by bringing the maximum speed higher than that of runner one, the remaining area is larger, and therefore more distance is covered by runner two to the point where they finish at the same time.

Therefore runner two has the highest maximum speed. Hope that makes sense.

3. Feb 13, 2013

### lorenamtimas

I think I get it. In regards to looking at what the graph may look like for this situation runner two has not a steeper slope but a higher one to compensate for the time it took for him/her to reach maximum speed. Do I have that right DarklightA?