Question

If a single constant force acts on an object that moves on a straight line, the object's velocity is a linear function of time. The equation v = v_i + at gives its velocity v as a function of time, where a is its constant acceleration. What if velocity is instead a linear function of position? Assume that as a particular object moves through a resistive medium, its speed decreases as described by the equation v = v_i - kx, where k is a constant coefficient and x is the position of the object. Find the law describing the total force acting on this object. (Enter an expression for the magnitude of the total force. Use the following as necessary: m, k, and v.)

Answer 1

Answer:

$F=mkv$

Explanation:

Given that

$v = v_i - kx$

We know that acceleration a given as

$a=\dfrac{dv}{dt}$

$v = v_i - kx$

$\dfrac{dv}{dt}=\dfrac{dv_i}{dt}-k\dfrac{dx}{dt}$

$\dfrac{dv}{dt}=0-k\dfrac{dx}{dt}$

We know that

$F=m\dfrac{dv}{dt}$

$F=-mk\dfrac{dx}{dt}$

$F=-mkv$

So the magnitude of force F

$F=mkv$