Careful measurements have been made of Olympic sprinters in...

Careful measurements have been made of Olympic sprinters in the 100-meter dash. A simple but reasonably accurate model is that a sprinter accelerates at $3.6 \mathrm{~m} / \mathrm{s}^2$ for $3 \frac{1}{3} \mathrm{~s}$, then runs at constant velocity to the finish line. a. What is the race time for a sprinter who follows this model? b. A sprinter could run a faster race by accelerating faster at the beginning, thus reaching top speed sooner. If a sprinter's top speed is the same as in part a, what acceleration would he need to run the 100 -meter dash in $9.9 \mathrm{~s}$ ? c. By what percent did the sprinter need to increase his acceleration in order to decrease his time by $1 \%$ ?

Careful measurements have been made of Olympic sprinters in the 100-meter dash. A simple but reasonably accurate model is that a sprinter accelerates at $3.6 \mathrm{~m} / \mathrm{s}^2$ for $3 \frac{1}{3} \mathrm{~s}$, then runs at constant velocity to the finish line.
a. What is the race time for a sprinter who follows this model?
b. A sprinter could run a faster race by accelerating faster at the beginning, thus reaching top speed sooner. If a sprinter's top speed is the same as in part a, what acceleration would he need to run the 100 -meter dash in $9.9 \mathrm{~s}$ ?
c. By what percent did the sprinter need to increase his acceleration in order to decrease his time by $1 \%$ ?

Key Concepts

Uniform Acceleration

This concept involves the motion of an object when it increases its velocity at a constant rate over time. The distance covered during uniform acceleration can be calculated using the equation for displacement in uniformly accelerated motion, which is essential for understanding how the sprinter's speed builds up during the initial phase of the dash.

Constant Velocity Motion

After the acceleration phase, the sprinter reaches a constant top speed and covers the remaining distance. In this phase, the distance traveled is the product of the constant velocity and the time, making it important to separate the motion into distinct segments for analysis.

Piecewise Motion Analysis

This concept refers to the breakdown of the motion into separate segments, each governed by different conditions—in this case, the accelerating phase and the constant velocity phase. Analyzing each piece individually allows for the application of appropriate kinematic equations to model the overall motion accurately.

Kinematics Equations

Kinematics equations relate variables such as displacement, velocity, acceleration, and time. These equations are vital in solving for unknown quantities, such as the race time or required acceleration, by connecting the dynamics of the accelerated motion with the constant velocity motion.

Percentage Change Calculation

This involves determining the relative change in a particular variable—in this instance, the increase in acceleration needed to decrease the race time by a certain percentage. Calculating percentage changes is important for understanding improvements or adjustments in performance metrics in various physics and engineering contexts.

Transcript

00:01 We're told that we're measuring an olympic sprinter in the 100 meter dash.

00:05 A reasonable model we're told is that the sprinter accelerates at 3 .6 meters per second squared for 3 and a third seconds, and then runs at a constant velocity to the finish line.

00:21 So, constant velocity afterwards, so in the second part, we have a is zero, and the total distance they run is 100 meters, of course.

00:32 So, again, i'm going to break the displacement down into the two phases, one where they're accelerating and one where they're running a constant velocity.

00:41 So in the acceleration phase, we know what at t equals zero, we're assuming that obviously they didn't cheat and they're starting with the zero velocity and they're starting at the starting line.

00:54 And so afterwards, we have this expression here.

00:58 But we also know that that a2 is zero.

01:04 So this term drops out.

01:06 And so we get v1, whatever that is the velocity at from the first, the initial velocity in the second period of the motion.

01:19 So that's v1 is actually x at t equals 3 in a third.

01:26 And then plus x1, which is the displacement at 3 in a third.

01:30 Seconds.

01:31 So again, we have these equations here.

01:39 And so we know x1, we can figure that out.

01:42 Again, is the displacement at 3 in a third seconds...