PLEASE ELLABORATE AS MUCH AS POSSIBLE. THANK YOU!!! Introduction The equation de

Question

PLEASE ELLABORATE AS MUCH AS POSSIBLE. THANK YOU!!!

Introduction The equation describing projectile motion with air resistance is: Solving this equation requires that you know 4 initial conditions (initial coordinates and 2 initial velocity components), the acceleration of gravity (not a big deal, but what if you want to compare situations on earth and on Mars?), the mass, and the drag coefficient. That's 7 numbers. What if we could eliminate some of them? We know from calculus that if you do a change of units via a rescaling of the variables, t' = const * t, the derivatives change as follows: This is like replacing a time measured in minutes with a time measured in seconds, so that the new time measurement is 60 times the old time measurement. Meanwhile, if we change units for the variable x, so that x' = const * x, the derivatives get multiplied by the constant. Can we redefine the velocity and time in such a way that the equations become simpler? Assignment Pick units of length and time such that two conditions are satisfied: The terminal velocity is 1. The equation for y can be written as: You don't necessarily have to make mg = 1; you could also change units in such a way that all of the terms on both sides of the equation have a factor of mg that cancels out.

Explanation / Answer

Terminal velocity ios given by vt = mg/c = 1 According to requirements

Also, d2y/dt2 = -c Vy sqrt(Vx2 + Vy2) - mg ....eqn 1

Now we make y' = k1 X y and t' = K2 X t

hence, d2y' / dt'2 = K1 / (K2 X K2)

multiplying by this factor in equation 1 , and equtating the terms with that given in question , we get:

c X K1 / (K2 X K2) = 1 and mg X K1 / (K2 X K2) = 1

Since we know that mg / c = 1

Hence Put K1 = c and K2 = c ANSWER

y' = c X y and t' = c X t

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