Question

how to calculate acceleration from velocity-displacement graph?. Its from kinematics

Answer 1

How to calculate acceleration from velocity-displacement graph? It's from kinematics.

Acceleration is the rate at which the velocity of a body changes with time:

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

We can multiply it by $1 = \frac{dx}{dx}$:

$$a = \frac{dv}{dt} \frac{dx}{dx} = \frac{dv}{dx} \cdot \frac{dx}{dt}$$

But by definition velocity equals:

$$v = \frac{dx}{dt}$$

Therefore:

$$a = \frac{dv}{dx} \cdot v$$

Therefore, acceleration equals slope on velocity-displacement graph multiplied by velocity.

For example, we have velocity-displacement graph and want to find acceleration in point $(s_0, v(s_0))$:

Slope in this point equals

$$\frac {\Delta v}{\Delta s} = \frac {d v (s)}{d s} \vert_ {s = s _ {0}}$$

And velocity equals $v(s_0)$ . Therefore, acceleration in this point:

$$a (s _ {0}) = \frac {\Delta v}{\Delta s} v _ {0} = \frac {d v (s)}{d s} | _ {s = s _ {0}} v (s _ {0}) = k * v (s _ {0})$$

where $k = \frac{dv(s)}{ds} \mid_{s=s_0} -$ slope in point $(s_0, v(s_0))$

Taking the slope of a v-t graph