## Related questions with answers

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 = vi + 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 = vi − 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.

Solution

VerifiedTo find the total (resultant) force acting on the body will cause it to accelerate, it will change the velocity of the body. This is just the statement of the Newton's second law.

$\begin{align*} F&=ma \end{align*}$

We have to find the acceleration of the body to find the force that act on it. The acceleration is the change of velocity in time. In terms of derivatives we have:

$\begin{align*} a&=\frac{dv}{dt} \end{align*}$

If we plug in the given function of velocity we get:

$\begin{align*} a&=\frac{d}{dt}(v_i-kx)\\ &=-k\frac{dx}{dt}\\ &=-kv \end{align*}$

Here we used the fact that the $v_i$ and $k$ are constants and they don't change with time and the fact that the change in position with time is velocity.

We can now express the total force as:

$\begin{align*} F&=-kvm \end{align*}$

But we know the expression of the velocity so:

$\begin{align*} F&=-km(v_i-kx)\\ &=\boxed{-kmv_i+mk^2x} \end{align*}$

The total force is a linear function of position.

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