Two-Dimensional Motion and Projectile Motion

• Two-dimensional motion involves movement in a plane, analysis of horizontal and vertical components of motion needed

Vector Components

• Vectors, like force or velocity, can be resolved into horizontal (x) and vertical (y) components using trigonometric functions
• If a vector V has a magnitude V and direction θ relative to the x-axis:
• Vx = V cos(θ)
• Vy = V sin(θ)
• These components are perpendicular and independent

Projectile Motion

• Projectile motion exemplifies two-dimensional motion under constant gravitational acceleration
• Ideal projectile motion assumptions include:
• Neglecting air resistance
• Constant gravitational acceleration (g) downwards
• The horizontal motion has constant velocity (ax = 0)
• The vertical motion has constant acceleration (ay = -g)
• Key equations for projectile motion include:
• Horizontal displacement: x = v0x * t
• Vertical displacement: y = v0y * t - (1/2) * g * t^2
• Vertical velocity: vy = v0y - g * t
• Time of flight (T) is the time it takes for the projectile to return to its initial vertical position (y = 0)
• Maximum height (H) is reached when the vertical velocity is zero (vy = 0)
• Range (R) is the horizontal distance traveled when the projectile returns to its initial vertical position
• Projectile launched from and landing on the same horizontal level:
• T = (2 * v0y) / g
• H = (v0y^2) / (2 * g)
• R = (v0^2 * sin(2θ)) / g
• The maximum range occurs when the launch angle θ = 45°

Relative Motion

• An object's velocity depends on the frame of reference from which it is observed
• If object A has velocity vA relative to frame F, and frame F has velocity vF relative to frame G, then the velocity of object A relative to frame G is: vAG = vA + vF, via vector addition
• In two dimensions, relative velocities must be added as vectors, considering both magnitude and direction

Force in Two Dimensions

• Forces are vectors that can be resolved into components
• The net force on an object is the vector sum of all individual forces
• Newton's Second Law (F = ma) applies in each dimension separately:
• ΣFx = m * ax
• ΣFy = m * ay
• Free-body diagrams are essential for visualizing forces on an object

Inclined Planes

• An inclined plane is a surface tilted at an angle to the horizontal
• Coordinate system choice for analyzing forces on an object on an inclined plane includes: x-axis parallel to the plane and the y-axis perpendicular to it
• Gravitational force (mg) resolves into components parallel (mg sinθ) and perpendicular (mg cosθ) to the plane
• The normal force (N) is equal in magnitude and opposite in direction to the perpendicular component of gravity: N = mg cosθ
• The force of friction (f) opposes the motion of the object along the plane, with f ≤ μN, where μ is the coefficient of friction

Circular Motion

• Uniform circular motion is movement in a circular path at a constant speed
• Velocity isn't constant because direction is always changing, even when speed is constant
• Centripetal acceleration (ac) is directed towards the center of the circle: ac = v^2 / r
• Centripetal force (Fc) is the net force causing centripetal acceleration: Fc = m * ac = m * (v^2 / r)
• In circular motion, speed (v) relates to radius (r) and period (T): v = 2πr / T
• The period (T) is the time it takes to complete one full revolution

Explores two-dimensional motion, focusing on vector components and projectile motion. Covers resolving vectors into horizontal and vertical components using trigonometry. Discusses projectile motion under constant gravitational acceleration, neglecting air resistance, and provides key equations for displacement and velocity.