Plane Curvilinear Motion - Choosing a Coordinate System

 

Rectanglar coordinates

 

The rectangular coordinate system is useful for straight line motion. It is also useful if the x and y component of the body’s acceleration are independent (e.g. projectile motion).

 

• Position: r = x i + y j
• Velocity: v = (dx/dt) i + (dy/dt) j
• Acceleration: a = (d ²x/dt ²) i + (d ²y/dt ²) j

 

n-t coordinates

 

The normal and tangential coordinate system is useful when a particle’s path is curved and information such as the path's radius of curvature or the speed and acceleration along the path are given.

 

• Velocity: v = v e_t
• Acceleration: a = (dv/dt) e_t + (v²/ρ) e_n

 

Polar coordinates

 

The polar coordinate system is useful when a particle’s path is curved especially if the path is circular. It is also useful when angular velocities and accelerations are given.

 

• Position: r = r e_r
• Velocity: v = (dr/dt) e_r + r(dθ/dt) e_θ
• Acceleration: a = ((d ²r/dt ²) - r(dθ/dt)²)e_r + (r(d ²θ/dt ²) + 2(dr/dt)(dθ/dt))e_θ