Susskind Lecture 2 - Motion
Lecture 2 replaces the discrete update arrows of Susskind Lecture 1 - Classical Mechanics with smooth motion. A particle’s trajectory is no longer a sequence of jumps; it is a function of time.
Plain-language version: if you know where something is as a function of time, calculus tells you how fast it is moving and how that motion is changing.
From position to velocity to acceleration
For a point particle in three-dimensional space, position is a vector-valued function:
r(t) = (x(t), y(t), z(t))Velocity is the time derivative of position:
v(t) = dr/dt
v_i = dx_i/dt = x_dot_iSpeed is the magnitude of velocity, not the velocity vector itself:
speed = |v| = sqrt(v_x^2 + v_y^2 + v_z^2)Acceleration is the time derivative of velocity, or the second time derivative of position:
a(t) = dv/dt = d^2r/dt^2This is the kinematic chain behind Kinematic: position → velocity → acceleration.
Constant acceleration
For vertical motion with constant downward acceleration, choose z upward. A typical trajectory is:
z(t) = z_0 + v_0 t - (1/2) g t^2
v_z(t) = v_0 - g t
a_z(t) = -gThe sign is not decoration. It encodes the direction of acceleration relative to the chosen axis.
Simple harmonic motion
An oscillating particle can be represented by a sine or cosine:
x(t) = A cos(ωt)
v(t) = -Aω sin(ωt)
a(t) = -Aω^2 cos(ωt) = -ω^2 x(t)The important physical statement is the last one: acceleration points back toward equilibrium. Position and acceleration are 180 degrees out of phase; position and velocity are 90 degrees out of phase.
The period of one full oscillation is:
T_period = 2π/ωUniform circular motion
A particle moving counterclockwise in a circle of radius R can be written:
x(t) = R cos(ωt)
y(t) = R sin(ωt)Differentiating gives a velocity tangent to the circle and acceleration toward the center:
|v| = Rω
a = -ω^2 r
|a| = Rω^2This is an early preview of a major theme: acceleration does not have to mean speeding up. In circular motion the speed is constant, but the direction of velocity changes, so the acceleration is nonzero.
Why Susskind spends time on calculus
Later lectures use derivatives everywhere:
- Susskind Lecture 3 - Classical Mechanics uses acceleration in Newton’s second law.
- Susskind Lecture 6 - Principle of Least Action varies paths and differentiates the Lagrangian.
- Susskind Lecture 8 - Hamiltonian Mechanics rewrites motion as first-order equations in phase space.
Lecture 2 is therefore not just a math interlude; it installs the language of continuous dynamics.
Common pitfalls
- Confusing velocity with speed. Velocity has direction; speed is only magnitude.
- Thinking acceleration means “going faster.” Acceleration means the velocity vector changes.
- Dropping signs in constant-acceleration problems. The sign tells you which way the acceleration points.
- Forgetting the chain rule when differentiating sin(ωt), cos(ωt), or any nested time function.
- Thinking circular motion has zero acceleration because speed is constant. The acceleration is inward.