Sage · Universal Tutor (preview)

Session · Mock data · No agent yet

Teach me kinematics

Kinematics is the branch of mechanics that describes how objects move — without yet asking why they move. We track three quantities through time: position, velocity, and acceleration. Each is a vector. Each is the time-derivative of the previous.

v(t) = \frac{dx}{dt}, \qquad a(t) = \frac{dv}{dt} = \frac{d^2 x}{dt^2}

Key idea

Velocity is the slope of position-vs-time. Acceleration is the slope of velocity-vs-time. The whole subject reduces to reading slopes off graphs.

Let's check: a particle moves with $x(t) = 3t^2 + 2t$ . What's its velocity at $t = 4$ ?

v(t) = \frac{dx}{dt} = 6t + 2 \quad \Rightarrow \quad v(4) = 26 \;\text{m/s}

Kinematic equations · constant acceleration

name	equation	missing
velocity-time	v = v₀ + at	x
position-time	x = x₀ + v₀t + ½at²	v
velocity-position	v² = v₀² + 2a(x − x₀)	t

Wikipedia · Looked up by Sage

Kinematics — Wikipedia

In physics, kinematics is the subfield of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move…

python · numerical example

import numpy as np
t = np.linspace(0, 5, 50)
x = 3*t**2 + 2*t
v = np.gradient(x, t)
print(f"v(4) ≈ {v[40]:.2f} m/s") # 26.00 m/s

What we just covered

Position, velocity, acceleration as time-derivative chain
Reading kinematic quantities off graphs (slope)
Computing v(t) from x(t) by differentiation

Sage is waiting for your next move…

Plan

What kinematics describes

Position, velocity, acceleration

Differentiation as the spine

Constant-acceleration equations

Worked problems

Quick quiz

Diagrams

📈 Position-time graph

📉 Velocity-time graph

References

📚 Kinematics — Wikipedia

      Velocity is the slope of position-vs-time. Acceleration is the slope of velocity-vs-time.
      Wait, why does the chain of derivatives stop at acceleration?
      Good question. Higher derivatives — jerk, snap, crackle, pop — exist, but rarely matter for textbook problems. Want to dig in?
    

What do you want to learn?

Continue