Projectile Motion: Analysis, Relationships & Air

• Projectile motion can be analyzed and predicted using models.

Projectile Motion Analysis

• Projectiles move with horizontal and vertical components.
• Constant vertical acceleration is due to gravity.
• Zero air resistance is assumed when modeling projectile motion.

Projectile Motion Relationships

• Quantitatively derive relationships between variables.
• Variables include initial velocity, launch angle, maximum height, launch height, time of flight, final velocity, and horizontal range of the projectile (ACSPH099).
• Practical investigations validate derived relationships.
• Problems are solved, and models/quantitative predictions are created using motion equations for uniformly accelerated and constant rectilinear motion.

Falling Through Air

• Weight describes the attractive force that the Earth exerts on objects within its gravitational field and is calculated by w = mg, where m is the object's mass and g is the acceleration due to gravity.
• Objects fall toward the Earth, but rates vary due to air resistance.
• Galileo suggested that the mass of an object did not define the rate at which an object fell, but the combined effect of the objects weight and resisting force.
• Air resistance, or drag (Fdrag), opposes an object's motion through the air.
• The net force acting on an object falling freely under the influence of gravity is equal to the vector sum of its weight and the drag acting upon it: Fnet = w + Fdrag, or ma = w + Fdrag
• a = (w + Fdrag)/m, so the rate at which an object accelerates as it falls towards the ground does not depend entirely on the object's mass or weight.

Air Resistance

• Drag is caused by the collision of a falling object with air particles, slowing its descent with greater collisions resulting in greater drag.
• The drag force is directly proportional to the object's cross-section perpendicular to motion direction, the density of the air, and the square of the object's speed.
• An object at 10 m/s experiences four times the drag compared to 5 m/s.
• Drag is also affected by the object's shape, smoothness, and turbulence.

Falling in a Vaccum

• Apollo 15 astronaut David Scott dropped a feather and hammer on the moons surface, and they fell at the same rate.
• Almost no drag in a vacuum
• Forces acting on both objects was gravity (Fnet = w) therefore ma = mg
• Acceleration was only affected by gravity and not mass.
• In a vacuum, all objects fall at the same rate, regardless of mass and size.

Projectile Motion Trajectory

• Projectiles include launched objects like basketballs, trapeze artists, and dropped packages.
• Trajectories follow a parabolic path, excluding straight up/down motion or guided missiles.
• Air resistance, object spinning, or wind can cause trajectory deviations.

Understanding Projectile Motion

• Galileo noted that projectile motions are separate and independent.
• Vertical motion is subject to gravity, while horizontal motion experiences no acceleration.
• Use the y-axis for vertical and the x-axis for horizontal motion.
• Perpendicular motions can be treated separately.

Projectile Motion and Velocity

• Projectiles are set in motion with velocity.
• A ball thrown directly upwards experiences only gravity (neglecting air resistance) and accelerates downwards.
• The ball slows down, halts, and then speeds up downwards, returning with the same speed it was projected with.
• The acceleration of the ball is a constant 9.8 m/s² downwards throughout the motion.

Horizontally Launched Projectiles

• An object launched horizontally follows a semi-parabolic trajectory with motion in two dimensions- horizontal and vertical.
• Horizontal components : Vx
• Vertical Components : Vy
• Gravity is the only continuous non-horizontal force so Vx remains constant throughout the flight.
• The vertical velocity profile is equal to an object dropped vertically, where vy = uy + gt
• Vertical Displacement : y = uyt + (1/2)gt²
• Without acceleration, horizontal displacement is x = vxt
• Parabolic motion is made by constant horizontal velocity and a uniformly changing vertical velocity.

Modelling Projectile Motion

• The vertical part of the motion determines the total time.
• Calculate range as horizontal displacement.

Projectiles at an angle

• Initial velocity resolved into horizontal and vertical for analysis
• ux = u cos θ
• uy = u sin 0
• Horizontal will remain the same (vx = ux.)
• Vertical changes in much the same manner as thrown objects.

Symmetrical Parabolic Motion

• Projectiles fired on a level surface are considered to have symmetrical trajectories.
• They reduce to a minimum at the peak, and once the projectile falls from its peak, its velocity increases again until, at the end of the trajectory, it has the same value as the initial velocity and even the same angle to the horizontal, although now it is directed below the horizontal.
• The maximum height has been reached halfway through the total time of flight.
• The bullet has reached half of its range.

Calculating Projectile Motion

• The time in flight connects the separate vertical and horizontal motions.
• Draw a diagram, separate motion into vertical/horizontal, resolve initial velocity.

Asymmetric Trajectories

• Projectiles are launched at a point above the horizontal plane.

Real World projectile motion

• Air resistance is ignored in motion analysis as it depends on various factors.
• Air resistance slows vertical and horizontal.
• Trajectory is therefore distorted.