Derivatives in Kinematics

Welcome! In this lesson, we'll be talking about how derivatives are applicable in kinematics. But first, a refresher on what a derivative is.

Alright!

You know what average velocity is, $$v_{av} = \frac{\Delta x}{\Delta t} = \frac{x_2 - x_1}{t_2 - t_1}$$

As always, this formula doesn't answer all our questions about velocity. For example, say you're driving on a road, and you want to know what your velocity is at a certain point in time. Well, certainly your speedometer can do this, but what if you're given your position as a function of time? Well, this calls for the instantaneous velocity, the velocity at a specific point. If you've studied calculus, you know that the velocity function is just the derivative of the position function with respect to time:

$$v(t) = \lim_{\Delta t \rightarrow 0}{\frac{\Delta x}{\Delta t}} = \frac{dx}{dt}$$

Similarly, we can find the acceleration of the car from the velocity function!

$$a(t) = \frac{dv}{dt}$$

Putting this together, we get:

$$a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2}$$

A body moves on a line. Its position is a function of time:

$x(t) = 5t^3 + t$

Let's find the velocity!

$v(t) = 15t^2 +1$

And now the acceleration!

$a(t) = 30t$

Suppose a drag force $F_{drag}(t) = -2t$ is exerted on the body during the whole motion. Find the equation of the force a girl should exert in order the body to move according to the above motion equations.

Assumptions:

Mass of the body = 1kg

Only these 2 forces (drag and girl-force) are exerted on the body.

All Units are S.I.

$F_{girl} = 32t$

Well Done!