Foundations of Projectile Motion A projectile is launched into a plane with its path described as a parabola. The motion unfolds in two dimensions, with horizontal and vertical positions changing over time. The scenario sets up the framework for analyzing movement using vector components and gravitational influence.
Decomposing Velocity Components The initial velocity is split into horizontal and vertical parts using trigonometric relations, with the horizontal component as u cosθ and the vertical as u sinθ. The horizontal motion remains uniform due to zero acceleration, while the vertical component is affected by gravity. This decomposition clarifies the distinct behaviors in each direction during a projectile’s flight.
Deriving Time of Flight, Range, and Maximum Height Setting the vertical displacement to zero allows derivation of the time of flight as 2u sinθ divided by gravity. Multiplying the horizontal velocity by this time yields the range, expressed as u² sin2θ divided by gravity. Meanwhile, the maximum height is determined from the vertical motion to be u² sin²θ divided by 2g.
Establishing the Trajectory Equation and Complementary Angle Symmetry Eliminating time from the equations leads to the trajectory relationship y = x tanθ minus (g x²)/(2u² cos²θ), which captures the parabolic path. The equation links horizontal and vertical displacements in a compact quadratic form. It further shows that two angles adding up to 90° produce the same range when launched with the same speed.
Apex Dynamics and Momentum Considerations At the highest point of its trajectory, the projectile’s vertical velocity drops to zero, leaving only the constant horizontal component. The time to reach this apex is exactly half the total flight time, calculated as u sinθ by g. This moment of zero vertical motion highlights a key change in momentum, bridging theoretical equations with practical applications.