Solving for v(t) & x(t) in a Region of Resistive Force

[joemama69]

Homework Statement

an object moves to the right with a constant speed v, the ovbject then enters a region where the resistive force is -bv², find v(t) & x(t)

Homework Equations

The Attempt at a Solution

F_x = ma = -bv²

v = (ma/-b)^.5

how do i get t in there

 

[CompuChip]

You are treating "a" and "v" as independent now.
However, the acceleration is related to the velocity as a = dv/dt.
So your equation is actually:
m dv/dt = - b v²

You are going to have to solve a differential equation.

 

[tomcue]

To get t in the equation, you can use the relationship between velocity and distance, which is given by v = dx/dt. This means that the derivative of the position function x(t) gives you the velocity function v(t). So, you can rewrite the equation as:

ma = -bv^2
m(dx/dt) = -bv^2

Now, you can rearrange the equation to solve for v(t) in terms of x(t):

dx/dt = -(bv^2)/m
dx = -(bv^2)/m * dt
dx = -bv^2 * (1/m * dt)
dx = -bv^2 * (1/m) * dt

Integrating both sides with respect to t, you get:

∫dx = -∫bv^2 * (1/m) * dt
x = -(b/m) * ∫v^2 * dt

To solve for v(t), you can take the square root of both sides and rearrange the equation:

v(t) = √(m/b) * √(dx/dt)

So, now you have both v(t) and x(t) in terms of each other. You can plug in values for x(t) to find the corresponding values for v(t) and vice versa.