Projectile Motion

Optimal Angle for Maximum Range

Projectile motion describes the path of an object launched into the air, subject only to gravity. The angle that maximizes the horizontal distance (range) for a projectile launched from ground level is a classic result in physics.

• Key Point: The optimal angle for maximum range is 45° when air resistance is neglected.

• Equation: The range of a projectile is given by:

• Example: A ball thrown at 45° will travel farther than one thrown at 30° or 60° (assuming equal initial speeds).

Friction and Circular Motion

Static vs. Kinetic Friction

Friction is the force that resists the relative motion of two surfaces in contact. Static friction prevents motion, while kinetic friction acts during motion.

• Key Point: The coefficient of static friction is usually higher than that of kinetic friction.

• Equation: Maximum static friction:

• Example: A car can travel faster around a curve if the tires have a higher coefficient of static friction with the road.

Normal Force on an Incline

The normal force is the perpendicular contact force exerted by a surface on an object resting on it. On an inclined plane, it is less than the object's weight.

• Equation:

• Example: A sled on a 20° slope experiences a normal force less than its weight.

Relative Motion and Velocity

Traveling in Perpendicular Directions

When two displacements are perpendicular, the total distance from the starting point is found using the Pythagorean theorem.

• Equation:

• Example: Traveling one mile north and one mile east results in a net displacement of miles northeast.

Forces and Tension

Tension in a Rope

Tension is the force transmitted through a rope, string, or cable when it is pulled tight by forces acting from opposite ends.

• Key Point: If two people pull with equal and opposite forces, the tension equals the force each applies.

• Example: If each person pulls with 100 N, the tension in the rope is 100 N.

Mass, Weight, and Apparent Weight

Apparent Weight in an Accelerating Elevator

Apparent weight is the normal force exerted on a person by the floor, which can differ from true weight if the elevator accelerates.

• Equation: (if accelerating upward)

• Example: A 60 kg person in an elevator accelerating upward at 2 m/s² feels an apparent weight of N.

Inclined Planes and Acceleration

Acceleration Down a Frictionless Ramp

On a frictionless incline, the acceleration is determined by the component of gravity along the ramp.

• Equation:

• Example: On a 30° ramp, m/s².

Circular Motion

Centripetal Acceleration

Objects moving in a circle at constant speed experience centripetal acceleration directed toward the center of the circle.

• Equation:

• Example: A car moving at 20 m/s around a curve of radius 100 m has m/s².

Collisions and Conservation of Momentum

Mid-Air Collisions

When two objects collide in mid-air, the law of conservation of momentum applies.

• Key Point: Both objects experience equal and opposite forces during the collision.

• Example: A bird colliding with a bee experiences the same magnitude of force as the bee does.

Work, Energy, and Power

Kinetic Energy and Work

Kinetic energy is the energy of motion, and work is the transfer of energy via force over a distance.

• Equation:

• Equation:

• Example: Pushing a block across a frictionless surface increases its kinetic energy.

Gravity and Free Fall

Free Fall and Acceleration Due to Gravity

Objects in free fall near Earth's surface accelerate downward at approximately 9.8 m/s², neglecting air resistance.

• Equation:

• Example: A rock dropped from 15 meters takes seconds to hit the ground.

Vectors and Vector Addition

Magnitude and Direction

Vectors have both magnitude and direction. The sum of two vectors depends on their relative directions.

• Equation: If and are perpendicular,

• Example: Adding a 3-unit vector east and a 4-unit vector north yields a resultant of 5 units northeast.

Tables: Forces on Stacked Objects

Force Distribution in Stacked Masses

When multiple objects are stacked, the force exerted by the table on each object depends on the total weight above it.

Additional info: The force on the bottom object is the sum of its own weight and the weights of all objects above it.

Extra Credit: Terminal Velocity and Fluid Resistance

Terminal Velocity in Fluids

Terminal velocity is the constant speed reached by an object when the force of gravity is balanced by the drag force of the fluid.

• Equation: For a sphere: (where is radius, is density)

• Example: Comparing two spheres of different radii and densities, the ratio of their terminal velocities can be calculated.

Summary Table: Key Equations