(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 $100-\mathrm{m}$ dash, what was his maximum speed for this race?

(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 $100-\mathrm{m}$ 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 Hi, in the given problem, the 100 meter is being finished by usain bolt in 9 .69 seconds.

00:09 So we can draw a velocity time curve for his motion in which we are representing the time over x -axis and velocity is being represented over y -axis.

00:26 Initially, he starts from rest.

00:29 So the curve will begin from the origin.

00:34 Now he accelerates for first three seconds.

00:39 So here this is accelerated motion for the three second.

00:44 Then this is three seconds from zero to final point when he stops accelerating.

00:54 This is the duration.

00:56 Then after which the velocity becomes constant.

00:58 So the remaining time out of the total time taken by him, which was 9 .69 seconds.

01:07 So the remaining time for the uniform motion will be 9 .69 minus 3 means this will be 6 .69 seconds.

01:17 So and here this is the maximum speed achieved by him, which he will keep as a constant after three seconds.

01:26 Now, the total distance covered by him will be given by the area covered by vt curve with the time axis.

01:47 And here this distance is given as 100 meter and this area will be divided into two parts, a1 and a2.

01:55 For a1, this is the area of right angle triangle, which is given as half, into base for base this is the time is this is three seconds and for height this is v which is missing then for a to this is the area of the rectangle which is the length and length is again time which is three six point six nine seconds into v so here it comes out to be hundred is equal to 1 .5 v plus 6 .69v.

02:31 If we add them, it will come out to be 8 .19v equals to 100.

02:42 So the first answer will be given by 100 by 8 .19 means the maximum speed achieved by usain bolt.

02:53 That is given as 12 .2.

02:56 Meter per second.

02:58 And this is one of the answer of this given problem.

03:02 Now we have to find the acceleration during this first part of its of his motion for which we will use first equation of motion...

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