(a) A world record was set for the men's 100-m dash in the...

(a) A world record was set for the men's 100-m dash in the 2008 Olympic Games in Beijing by Usain Bolt of Jamaica. Bolt “coasted” across the finish line with a time of 9.69 s. If we assume that Bolt accelerated for 3.00 s to reach his maximum speed, and maintained that speed for the rest of the race, calculate his maximum speed and his acceleration. (b) During the same Olympics, Bolt also set the world record in the 200-m dash with a time of 19.30 s. Using the same assumptions as for the 100-m dash, what was his maximum speed for this race?

Key Concepts

Uniform Acceleration

Uniform acceleration refers to a scenario where an object's velocity increases or decreases at a constant rate over time. This concept allows the use of specific kinematic equations that relate displacement, initial velocity, time, acceleration, and final velocity, making it essential for solving problems where acceleration is constant during a time interval.

Constant Velocity

Constant velocity describes motion in which an object moves at an unchanging speed and direction, meaning its acceleration is zero. When analyzing such motion, the distance traveled can be directly calculated by multiplying the constant speed by the time spent moving, simplifying part of the problem.

Piecewise Motion Analysis

Piecewise motion analysis involves breaking down a motion problem into distinct segments, each with its own characteristics, such as an acceleration phase followed by a constant speed phase. This method enables the appropriate application of different kinematic equations to each segment, providing a comprehensive solution to complex motion scenarios.

Kinematic Equations

Kinematic equations are mathematical formulas that relate displacement, velocity, acceleration, and time in motion problems. They are instrumental in calculating unknown quantities in segments of both uniformly accelerated and constant velocity motion, thereby forming the backbone of problem-solving in introductory mechanics.

Transcript

00:01 So we can say for part a, first three seconds, we have that the maximum velocity is equaling the initial velocity, which we know to be zero at the beginning.

00:18 This would be plus the maximum acceleration times t of 3 .00 seconds.

00:24 And so the maximum velocity would be equaling 3 .00 times the maximum acceleration.

00:30 Now, the distance traveled within those three seconds, we can say x sub 1.

00:36 This would be equal to one half times the maximum acceleration times 3 .00 seconds squared.

00:45 And so this would be equalling one half times the maximum velocity divided by three multiplied by nine.

00:56 And this is equaling 1 .50 times the maximum velocity.

01:01 Now, because in the next 6 .69 seconds, bolt did not accelerate, so we can say x sub 2 is simply equalling 6 .69 times the maximum velocity, since he keeps this velocity for the next 6 .69 seconds.

01:19 We know that the total distance x sub 1 plus x sub 2 should equal 100 meters, because it's a 100 meter dash, and this is equal 1 .5 maximum.

01:32 1 .50 times the velocity max, multiplied by 6 .69 times a velocity max.

01:38 And so this would be equaling 7, so 8 .19 times the maximum velocity.

01:48 And so to solve for the maximum velocity, this would be 100 divided by 8 .19, 12 .2 meters per second.

01:58 And so the maximum acceleration is equaling 12 .2 meters per second divided by three, and this is equaling 4 .07 meters per second squared.

02:12 This would be our final answer for the maximum acceleration during those first three seconds.

02:17 That would be our answer for part a.

02:20 For part b, we have that here...

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