Derivatives of Parametric and Polar Curves - Theory & Concepts - Calculus Projectile Parametric

Learn how to find derivatives, slopes, and tangency angles for curves defined parametrically and in polar coordinates.

Parametric Trajectory: Projectile Motion

Explore how parametric derivatives govern vertical and horizontal velocity rates, constructing the instantaneous tangent slope vector $dy/dx$ .

Launch Velocity (

v_0

): 30 m/s

Launch Angle (

\theta

): 45°

Time (

t

): 2.16 / 4.32 s

0.00sPeak: 2.16sImpact: 4.32s

Parametric Derivatives

x(t) position:45.82 m

y(t) position:22.94 m

dx/dt (Horizontal rate):21.21 m/s

dy/dt (Vertical rate):0.02 m/s

Tangent Slope dy/dx:0.0011

Concavity d²y/dx²:-0.02180 m⁻¹

Dynamic Motion Path & Tangent Vector

Observation: Horizontal velocity

v_x = dx/dt

is constant throughout flight (ignoring air drag). Vertical velocity

v_y = dy/dt

decreases linearly from positive to negative due to gravity. The combined velocity vector is always exactly tangent to the curve, represented by the parametric derivative

dy/dx

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Parametric Trajectory: Projectile Motion

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