Who can differentiate this one?

[Fiasal teslla]

For a falling object, who can find the time in terms of the velocity for this journey( before reaching terminal velocity) with considering the resistive force?

 

[g_edgar]

Force equals mass times acceleration
Choose what resistive force to use, inverse velocity or inverse square velocity
Write down the differential equation
Solve it

 

[HallsofIvy]

I can"! But are you clear on the terminology? Your title asked "how can differentiate this one" implying you have a function to differentiate but that doesn't appear to be the case. You are just asking about setting up the dynamic equation.

As g edgar says, "Force equals mass times acceleration" so ma= m dv/dt= -g- f(v) where "f(v)" is the resistive force. That can be a very complicated function of the velocity depending on the situation. I do not agree with g edgar's "inverse" formulas. Typically, the faster something is going, the greater the drag, not the other way around. Normally, the drag is simplified to either -kv or -kv² where k is the constant of proportionallity and v is the speed.

 

[Fiasal teslla]

I can"! But are you clear on the terminology? Your title asked "how can differentiate this one" implying you have a function to differentiate but that doesn't appear to be the case. You are just asking about setting up the dynamic equation.

As g edgar says, "Force equals mass times acceleration" so ma= m dv/dt= -g- f(v) where "f(v)" is the resistive force. That can be a very complicated function of the velocity depending on the situation. I do not agree with g edgar's "inverse" formulas. Typically, the faster something is going, the greater the drag, not the other way around. Normally, the drag is simplified to either -kv or -kv² where k is the constant of proportionallity and v is the speed.

When I tried to solve it, I started with this equation
F-R=ma, where F= mg, snd R is the risitive force which equals bv (b is a constant)
So, mg-bv=ma
mg-bv=m dv/dt then we separate the variables to get the time in terms of the velocity

Am I correct with that?

 

[HallsofIvy]

Yes, in that case, with resistive force proportional to v, you get a fairly simple equation:m dv/dt= mg- bv, a separable equation. But be careful about signs. If you are taking "upward" to be positive, then it is m dv/dt= -mg- bv. Since drag always acts opposite to velocity, the coefficient of v is always negative.