Velocity & acceleration

Japanese school year: Math III

What you learn

This topic demonstrates calculating velocity and acceleration by taking the first and second derivatives of a position function with respect to time. It is an indispensable tool in classical mechanics and engineering simulations for describing motion. Students should be proficient in differentiating polynomial and trigonometric functions before taking this module.

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Key points

This formula calculates the instantaneous velocity $v$ of an object moving along a line. Differentiating position $x$ with respect to time $t$ gives both speed and direction.

v = \frac{dx}{dt}

This formula determines the acceleration $a$ , which measures how fast velocity changes. It is obtained by differentiating velocity $v$ once, or position $x$ twice, with respect to time $t$ .

a = \frac{dv}{dt} = \frac{d^2x}{dt^2}

This formula calculates the speed (magnitude $|\vec{v}|$ of velocity vector $\vec{v}$ ) of an object moving in a plane. Square both horizontal velocity $\frac{dx}{dt}$ and vertical velocity $\frac{dy}{dt}$ , sum them, and take the square root.

|\vec{v}| = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}

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Practice 1Coming soon