Two Dimensional Kinematics-Projectile Motion-Motion in 2-D

Explore two-dimensional motion and projectile motion by analyzing a ball thrown from the ground at angle theta, uncovering maximum height, range, and flight time, with independence of perpendicular directions.

Choose a coordinate system for projectile motion, ground as x and vertical as y. Include origin at throw point and note time of flight, maximum height, range, and trajectory equation.

Resolve projectile motion into x and y components under constant gravity, neglect air resistance, and apply independent perpendicular directions to derive velocity and displacement.

Explore two-dimensional projectile motion by analyzing time of flight, maximum height, and range through horizontal and vertical components. Derive the trajectory equation from initial velocity components and gravity.

Explore alternate methods to compute the maximum height in projectile motion by maximizing the y displacement using calculus and a quadratic approach, with initial velocity components and gravity.

Derive the horizontal range of a projectile using cartesian components, time of flight, and angle dependencies, including maximum range at 45 degrees and complementary angles.

Derive the trajectory equation for a projectile, showing path is a parabola and eliminating time to obtain y = x tan θ − g x^2 / (2 u^2 cos^2 θ).

Illustrates two-dimensional projectile motion with a 20 m/s throw at 53°, giving x = 12 t, y = 16 t − 5 t^2, flight time 3.2 s, range 38.4 m.

Analyze a two-dimensional projectile with initial velocity components; compute velocity vectors at 11 meters displacement and at 1 and 2.2 seconds; determine angles between total velocity and x and y components.

solve a two-wall projectile problem by computing the 14.4 meter spacing between walls of height 11 meters from a 1.2 second crossing interval, given horizontal velocity 12 m/s.

Analyze a two-dimensional projectile motion problem to determine the catcher's required speed to meet the projectile on the ground. Compute range, time of flight, and tangential and radial acceleration components.

Explore two-dimensional projectile motion with a horizontally thrown ball, where horizontal velocity stays constant, vertical motion accelerates under gravity, defining time of flight, range, and the independence of perpendicular motions.

this lecture analyzes a projectile launched from a 16 m tower at 20 m/s and 53°, yielding a 4 s flight, 48 m range, and 28.8 m max height.

Analyze a projectile from the mountain base that grazes the peak, using a modified trajectory and two ideas: range equals base length, and top lies on the trajectory (alpha, h).

Explore five projectile motion problems, deriving horizontal and vertical components, maximum height, time of flight, range, and velocity at a given height, including incline and wind effects.

Explore a projectile problem with multiple solution methods, linking concepts through dot products, tilted coordinates, and trajectory geometry to find when velocity is perpendicular to the initial velocity.

Projectile motion on an inclined plane uses coordinates along the plane and perpendicular to it to analyze time of flight, range on incline, and height perpendicular to the plane.

Analyze a projectile thrown perpendicular to an inclined plane, determine the plane range via time-of-flight and incline coordinates, and compare with a standard horizontal-coordinate approach.

Explore two incline-plane projectile problems: a vertical throw on a 30° incline and a horizontal launch from a 45° incline, using normal and tilted coordinates to compute the range.

Analyze a horizontally projected particle from a 10 m height onto a 45-degree incline, computing horizontal range, time of flight, and distance along the incline using normal and tilted coordinates.

Explore range on an incline for a projectile thrown perpendicular to a 30-degree plane, solved via normal and tilted coordinate methods, yielding a 40/3 meter range.

Determine the initial speed for a projectile launched at angle beta to just graze an inclined roof at alpha, using trajectory tangency and a rotated-axis maximum-height method.