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) is a classic result in introductory physics.

• Key Point: The optimal launch angle for maximum range on level ground (ignoring air resistance) is 45°.

• Equation: The range of a projectile launched at speed and angle is:

• Example: A ball thrown at 45° will travel farther than one thrown at 30° or 60° (with the same speed).

Friction and Circular Motion

Static vs. Kinetic Friction

Friction is the force that resists motion between two surfaces. 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 force:

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

Friction on Inclined Planes

When an object is on an inclined plane, friction opposes the component of gravity parallel to the surface.

• Key Point: The angle at which an object starts to slide is determined by the coefficient of static friction.

• Equation:

• Example: If , the table must be tilted to before an object starts sliding.

Relative Motion and Distance

Traveling in Perpendicular Directions

When moving in perpendicular directions, the total displacement is found using the Pythagorean theorem.

• Key Point: Traveling 1 mile north and 1 mile east results in a total displacement of miles.

• Equation:

• Example: Walking 3 miles north and 4 miles east gives miles.

Forces and Newton's Laws

Tension in a Rope

When two people pull on a rope with equal force, the tension equals the force applied by each person.

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

• Equation: (if forces are equal and opposite)

Normal Force on Inclined Planes

The normal force is the perpendicular contact force exerted by a surface.

• Key Point: On an incline,

• Example: For a 25 kg sled on a 20° slope:

Apparent Weight in Accelerating Elevators

Apparent weight changes when an elevator accelerates.

• Key Point: If the elevator accelerates upward, apparent weight increases:

• Example: A 60 kg person in an elevator accelerating upward at has

Uniform Circular Motion

Centripetal Acceleration

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

• Key Point:

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

Collisions and Conservation of Momentum

Mid-Air Collisions

When two objects collide in mid-air, momentum is conserved.

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

• Example: A bird colliding with a bee: both change velocity according to conservation of momentum.

Vector Addition and Properties

Magnitude of Vector Sums

Vectors can be added using the parallelogram rule or component-wise.

• Key Point: If and are perpendicular,

• Equation:

Gravity and Free Fall

Projectile Range and Height

The range and maximum height of a projectile depend on initial speed, angle, and gravity.

• Key Point: On the Moon, with lower gravity, projectiles travel farther and higher.

• Equation:

Tables: Comparison and Classification

Sample Table: Apparent Weight in Elevator

Sample Table: Friction on Inclined Planes

Additional info:

• Some questions reference air resistance proportional to velocity squared, which is typical for high Reynolds number flows.

• Terminal velocity is reached when the force of gravity equals the force of air resistance.

• Questions about vector addition and perpendicular vectors reinforce the importance of understanding vector properties in physics.