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Solve 1D/2D kinematics, projectile trajectories, Newton's laws of motion (F=ma), work, kinetic/potential energy, momentum conservation, and harmonic oscillations with exact SI physical constants.
v_0 at angle heta above horizontal: v_{0x} = v_0 cos( heta) (constant horizontal velocity) and v_{0y} = v_0 sin( heta) (vertical initial velocity).t_{ ext{apex}} = rac{v_{0y}}{g}; Total flight duration (flat ground): T = rac{2 v_{0y}}{g}; Maximum Apex Height: H_{ ext{max}} = rac{v_{0y}^2}{2g}; Total Horizontal Range: R = v_{0x} imes T = rac{v_0^2 sin(2 heta)}{g}. ec{F}_{ ext{net}} = m ec{a}. On an inclined plane of angle alpha: parallel sliding force F_{parallel} = m g sin(alpha), normal force F_{perp} = m g cos(alpha), friction f_k = mu_k F_{perp}.E_{ ext{tot}} = KE + PE = rac{1}{2}m v^2 + m g h = ext{constant}. Impact velocity upon returning to launch height: v_{ ext{impact}} = sqrt{v_{0x}^2 + v_{0y}^2} = v_0.In classical Newtonian physics neglecting air resistance, the only force acting on a free projectile in flight is gravity, which acts strictly in the vertical downward direction (ay = -g). Because there is zero net force acting horizontally (Fx = 0), horizontal acceleration is zero (ax = 0), keeping horizontal velocity vx constant throughout flight.
The four kinematic equations describe motion under constant acceleration: 1) v = u + at (velocity-time); 2) s = ut + 0.5at^2 (displacement-time); 3) v^2 = u^2 + 2as (timeless velocity-displacement); 4) s = 0.5(u + v)t (average velocity displacement). Each equation omits exactly one of the five kinematic variables (s, u, v, a, t), allowing you to solve any system if three parameters are known.
The Work-Energy Theorem states that the net work done on an object by all forces equals its change in kinetic energy: W_net = ΔKE = 0.5*m*v_final^2 - 0.5*m*v_initial^2. This allows you to solve for final speeds without integrating complex accelerations over time.
In real atmospheres, fluid air resistance exerts a quadratic drag force F_d = 0.5 * rho * v^2 * C_d * A opposite to velocity. This creates an asymmetric trajectory where the descent angle is steeper than the launch angle, reduces maximum range, lowers apex height, and shifts the optimal launch angle below 45° (typically 35°-42° for athletic projectiles).