College Physics| Openstax| Problem 2.40

(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?

 SOLUTION:

Part a

There are two parts to the race and must be treated separately since acceleration is not uniform over the race. We will divide the race into x_1 (while accelerating) and x_2 (with constant speed), where x_1+x_2=100\:m.

When the speed is constant, t=6.69s, so

x_2=vt=\left(6.69\:s\right)v=100m-x_1

When accelerating, t=3.00s, so

x_1=\frac{1}{2}at^2=\frac{1}{2}vt=\left(1.50\:s\right)v

Plugging x_1 into the previous equation, we get the maximum speed: 

100m-\left(1.50s\right)v=\left(6.69s\right)v

v=\frac{100m}{6.69s+1.50s}

v=12.2\:m/s

Therefore, his acceleration was

a=\frac{v}{t}

a=\frac{12.2\:m/s}{3.00\:s}

a=4.07\:m/s^2

Part b

Similar to part (a), we can plug in the different values for time and total distance: 

200m-\left(1.50\:s\right)v=\left(16.30\:s\right)v

v=\frac{200\:m}{16.30\:s+1.50\:s}

v=11.2\:m/s

 

 

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