Question

l Review Careful measurements have been made of Olympic sprinters in the 100- meter dash. A quite realistic model is that the sprinter's velocity is given by a(1-eR) where t is in s, vx is in m/s, and the constants a and b are characteristic of the sprinter. Sprinter Carl Lewis's run at the 1987 World Championships is modeled with a -11.81m/s and b -0.6887s ,

Answer 1

Part D :

We know that , velocity is defined as the time derivative of displacement. i.e.

  

So,

Given,

And for Carl Lewis

a = 11.81 m/s

b = 0.6887 s^-1

^So,   

^or, ..................................(1)

Where, C is integration constant which can be calculated using initial conditions.

The initial condition is : at time t = 0 second the position of the sprinters was at x = 0 m. Putting these values in equation (1), we get

  

or,   

or,   

Putting the value of C in equation (1), we get

.........................................(2)

This is the required expression for distance travelled at time t.

Part E :

Putting the values of a and b in equation (2), we get

or,

For x = 100 m , we can write

  

or,

Dividing both sides by 11.81, we get

   ............................(3)

Now using trial and error method :

Let t = 10 seconds and put it into the RHS (right hand side) of equation (3) ,we get

or,   

Which is greater than the LHS (left hand side) of the equation (3) i.e. 9.919 .

let t = 9.9 second then

Which is nearly equals to LHS of the equation (3).

Let t = 9.918 second then

  

R.H.S 9.918 + 1.452e一0.6887×9.918ー9.91 1.452 × 0.001080 9.9195

Which is equals to the LHS of equation (3) .

Hence, the Lewis needed 9.918 second (or 9.92 second) to sprint 100 m .

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