Lecture 4 1d to 2d

Transition from 1D to Multidimensional Motion

• Focus on transitioning from one-dimensional motion along x-axis to multiple dimensions (2D or 3D).

• Target: Simple cases for solving Newton's second law for multidimensional motion.

Example: Motion in a Plane

• Consider 2D problems involving two particles (nucleus and classical electron in a hydrogen atom).

• Approach classical mechanics, although real scenario would be quantum mechanical.

• Reduce two particles to one using coordinate transformations.

Coordinate Transformation

• Transform from Cartesian coordinates to Polar coordinates:

  • Cartesian: Requires Newton's second law.

  • Polar: Simplifies the problem in the context of motion.

Mathematical Interlude

Position in Cartesian Coordinates

• Define a point P in 2D space with coordinates (x, y).

• Origin for convenience positioned for calculations.

• Distance from origin to point P:

  • Magnitude, r = √(x² + y²) (scalar quantity).

Position Vector

• Position is specified by both x and y components:

  • Vector notation: r = x î + y ĵ.

• î and ĵ are unit vectors in x and y directions respectively:

  • î represents length 1 along the x-axis.

  • ĵ represents length 1 along the y-axis.

Polar Coordinates

• Position can also be specified using Polar coordinates (r, θ):

  • r: distance from origin.

  • θ: angle with respect to the x-axis.

• Conversion relationships:

  • r = √(x² + y²)

  • θ = arctan(y/x)

Components from Polar Coordinates

• If distance r and angle θ are known:

  • x = r cos(θ)

  • y = r sin(θ)

Distances Between Particles in Cartesian Coordinates

Distance Between Particles A and B

• Positions of particles given as:

  • Particle A: coordinates (xa, ya)

  • Particle B: coordinates (xb, yb)

• Distance (d) calculation:

  • d = √[(xb - xa)² + (yb - ya)²].

Position Vector for Distance Between A and B

• Vector notation for distance between particles:

  • rab = (xb - xa) î + (yb - ya) ĵ.

Newton's Second Law in 2D

Forces in 2D Motion

• Consider a particle launched at angle θ with initial velocity v0 under gravity.

• Forces acting:

  • Force in y: -mg (downwards due to gravity).

  • Force in x: 0 (no horizontal force applied).

• Total force vector:

  • F = 0 î - mg ĵ.

Newton's Equations of Motion

• For x-direction:

  • F_x = m(dV_x/dt) = 0 ➔ V_x is constant.

• For y-direction:

  • F_y = m(dV_y/dt) = -mg ➔ dV_y/dt = -g.

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Solving Motion Equations

• For x-direction:

  • V_x = v0 cos(θ) ➔ X(t) = v0 cos(θ) t.

• For y-direction:

  • dV_y = -g dt ➔ V_y = v0 sin(θ) - gt.

Summary

• Motion in 2D requires understanding of Cartesian and Polar coordinates.

• Key transformations and vector calculations are foundational.

• Applications of Newton's second law are vital in both component analyses.