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I am trying to work out the velocity-time relationship for a vehicle under certain conditions. The 'certain conditions' are:

1. deceleration due to aerodynamic drag and rolling friction

2. acceleration due to an engine less drag and friction

now, i will address the deceleration first. I tried to do model this in two ways:

a) Equilibrium of forces giving: -D - f = ma (where D=drag, f=friction). putting in the known values, i get the value of acceleration in terms of velocity as:

a = -0.00075v^{2}-0.2943

i know that a=dv/dt, so re-arranging, i get: dt=dv/a as a=f(v)

Integrating this, i got:

v = 19.8 x tan (((t_{1}-t)/67.3) - tan^{-1}(v/19.8))

now, QUESTION: does this formulation make sense? and second, would the angle be in radians or degrees?

i put in values both in degrees and radians, but found the deceleration to be quite fast with radians and very slow with degrees. By fast and slow, i mean not in sync with practice. For example:

with radians, the vehicle goes from 10 m/s to 5m/s in 15s with a fairly constant gradient. Does it make sense?

b) with the basic conservation of energy equation, starting from:

KE_{1}- (P_{f}+ P_{D})xt = KE_{2}

Putting the values in, i got this relationship b/w time-velocity:

t = 666.7 x (100 - v^{2})/(v x (v^{2}+392.4))

the deceleration (v-t) curve i got was more moderate, with a markedly decreasing gradient

QUESTION: How do these two methods compare?

What am i doing wrong?

What else should i be doing?

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# Velocity-Time relationship for acceleration/deceleration

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