Terminal velocity of a bicycle under a constant force

[Imperial Sky]

Hello!

The total force F = m*a applied to the bicycle is F = F1 - a*v^2,
where F1 is the initial force applied to the bicycle, a is a constant and v is the velocity.
That means that the total force applied to the bicycle decrease as velocity increases, like a wind resistance decreases acceleration. I know how to find the terminal velocity (maximum velocity that a bicycle can reach under constant initial force), I have to set the total force equal to 0 and then find v.

But how can I present velocity v as a function of time so I can see how velocity on the graph approaches terminal velocity?

Sorry if I didn't explain it well, I wrote in intuitively. I am thankful if anyone can help me.

 

[berkeman]

Hello!

The total force F = m*a applied to the bicycle is F = F1 - a*v^2,
where F1 is the initial force applied to the bicycle, a is a constant and v is the velocity.
That means that the total force applied to the bicycle decrease as velocity increases, like a wind resistance decreases acceleration. I know how to find the terminal velocity (maximum velocity that a bicycle can reach under constant initial force), I have to set the total force equal to 0 and then find v.

But how can I present velocity v as a function of time so I can see how velocity on the graph approaches terminal velocity?

Sorry if I didn't explain it well, I wrote in intuitively. I am thankful if anyone can help me.

Welcome to the PF.

Thanks for re-posting this here in the Homework Help forums instead of the ME forum. Next time, though, please fill out the HH Template you are provided when starting a schoolwork thread. It helps to organize your equations and thoughts on the question.

Now to your question. Use F=ma to give you the acceleration as a function of time a(t) = F(t)/m, and then integrate that to get the velocity v(t). Does that help?

 

[haruspex]

Use F=ma to give you the acceleration as a function of time a(t) = F(t)/m, and then integrate that to get the velocity v(t). Does that help?

it's not quite that simple. v=v(t) will be on both sides of the equation (as dv/dt on the left). So the right hand side cannot be directly integrated. Instead, treat it as a differential equation in v and t.