# Relationship Between Velocities of Two Runners

• Drakkith
In summary, the conversation discussed the relation between the constant speed of two runners, Josie and Reginald, on a straight track. It was determined that the distance traveled is the integral of velocity over time, making the solution a simple geometric exercise. The correct answer is that Vj = 1/2 Vmax.
Drakkith
Mentor

## Homework Statement

Reginald is out for a morning jog, and during the course of his run on a straight track, he has a velocity that depends upon time as shown in the figure below. That is, he begins at rest, and ends at rest, peaking at a maximum velocity Vmax at an arbitrary time tmax. A second runner, Josie, runs throughout the time interval t = 0 to t = t f at a constant speed Vj, so that each has the same displacement during the time interval. Note: t f is NOT twice t max , but represents an arbitrary time. What is relation between Vj and tmax?

## Homework Equations

Position equation for constant acceleration: X=Xi+Vi+1/2AT2
Velocity Equation for constant acceleration: V=Vi+AT
Position Equation for constant velocity: X=Xi+VT

3. The Attempt at a Solution

Not sure what to do really, so I've just been trying things.
The only commonality between the two runners is the final position X and the time tf

For Josie, X = Vjtf

For Reginald:
Velocity:
From ti to tmax, Vmax = Atmax
From tmax to tf, since Vf = 0, the equation is: 0 = Vmax + Atf

Position:
From ti to tmax, initial displacement and velocity are zero: X1 = 1/2Atmax2
From tmax to tf: X = X1 + Vmaxtf + 1/2Atf2Since X = Vjtf, we can rewrite the above equation as: Vjtf = X1 + Vmaxtf + Atf2
Replacing X1 with its equation: Vjtf = 1/2Atmax2+ Vmaxtf + Atf2

That's about as far as I've gotten and I don't know if I'm even on the right track.

Distance traveled is the integral of velocity over time -- the area under the velocity versus time graph. Given that, the problem turns into a simple geometric exercise. One needn't bother with any equations at all.

jbriggs444 said:
Distance traveled is the integral of velocity over time -- the area under the velocity versus time graph. Given that, the problem turns into a simple geometric exercise. One needn't bother with any equations at all.

Thanks Jbriggs. Apparently the question was asking about Vmax, not Tmax, as Vj = 1/2 Vmax is the correct answer. (Which wouldn't have helped me prior to your post anyways, I still had no idea to look for the area under the graph)

Edit: Just so I don't look like an idiot, the question I posted was a literal copy and paste. I didn't just misread Vmax as Tmax.

Last edited:

## 1) What is the relationship between the velocities of two runners?

The relationship between the velocities of two runners is typically described as a comparison between their speeds or rates of motion. It is commonly measured in terms of distance covered per unit of time, such as miles per hour or meters per second.

## 2) How is velocity different from speed?

While velocity and speed are often used interchangeably, they are not exactly the same. Velocity is a vector quantity that includes both speed and direction, while speed is a scalar quantity that only measures the rate of motion.

## 3) What factors can affect the velocities of two runners?

There are several factors that can affect the velocities of two runners, including their physical abilities, training techniques, and environmental conditions such as wind resistance or terrain. Additionally, external factors such as motivation and strategy can also play a role.

## 4) How can the velocities of two runners be compared?

The velocities of two runners can be compared by measuring their speeds over a fixed distance or time interval. This can be done using a stopwatch or other timing device, as well as with the help of video analysis software or other advanced tools.

## 5) Is the relationship between the velocities of two runners always the same?

No, the relationship between the velocities of two runners can vary depending on the specific context and conditions. For example, if one runner is sprinting while the other is jogging, their velocities will likely be different. Additionally, factors such as fatigue or injuries can also impact the relationship between velocities.

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