Question

An object of mass m is moving in a straight line with velocity v and is slowing down from a force F = -kv where k is a constant. At time t = 0, the velocity is $v_{0}$. How can I derive an equation for the object's velocity as a function of t?

Answer 1

Answer:

The equation of the object's velocity in time is $v(t) = v_{o}\cdot e^{-\frac{k\cdot t}{m} }$.

Explanation:

By Newton's Laws of Motion, the equation of motion that represents the deceleration of the object is described by:

$\Sigma F = - k\cdot v = m\cdot \frac{dv}{dt}$ (1)

Where:

$k$ - Damping constant, in newton-second per meter.

$m$ - Mass, in kilograms.

$v$ - Velocity, in meters per second.

$\frac{dv}{dt}$ - Acceleration, in meters per square second.

Then, we modify (1) until the following ordinary differential equation with separable variables is found:

$-\frac{k}{m} \int \, dt = \int {\frac{dv}{v} }$ (2)

Then, we integrate the equation and find the following solution:

$-\frac{k}{m}\cdot (t-0) = \ln \frac{v}{v_{o}}$

$- \frac{k\cdot t}{m} = \ln \frac{v}{v_{o}}$

Finally, we clear the velocity in the solution of the differential equation is:

$v(t) = v_{o}\cdot e^{-\frac{k\cdot t}{m} }$

The equation of the object's velocity in time is $v(t) = v_{o}\cdot e^{-\frac{k\cdot t}{m} }$.