Motion in One and Two Dimensions

Projectile Motion

Projectile motion involves the movement of an object thrown or projected into the air, subject only to acceleration due to gravity. The horizontal and vertical motions are analyzed separately.

• Horizontal Range: The distance traveled by a projectile launched at an angle with initial speed is given by:

• Maximum Height:

• Time of Flight:

• Example: A projectile is shot from ground level at an angle above the horizontal. To find the range, use the above formula, assuming no air resistance.

Relative Motion

Relative motion problems involve comparing the motion of objects from different reference frames.

• Example: If Anna drives east and Maria drives northeast, the difference in their positions can be found using vector addition and the Pythagorean theorem.

Forces and Newton's Laws

Newton's Laws of Motion

Newton's laws describe the relationship between the motion of an object and the forces acting on it.

• First Law (Inertia): An object remains at rest or in uniform motion unless acted upon by a net external force.

• Second Law:

• Third Law: For every action, there is an equal and opposite reaction.

• Example: Calculating the tension in a rope supporting a mass in an elevator accelerating upward.

Friction

Friction is a force that opposes the motion of objects. The kinetic friction force is proportional to the normal force:

• Example: An object sliding down a ramp experiences friction proportional to its weight and the coefficient of friction.

Circular Motion and Gravity

Uniform Circular Motion

When an object moves in a circle at constant speed, it experiences a centripetal acceleration directed toward the center of the circle.

• Centripetal Force:

• Example: A car traveling around a circular track requires a frictional force to provide the necessary centripetal acceleration.

Gravity and Free Fall

Objects in free fall experience acceleration due to gravity, , which is approximately on Earth. On the Moon, is about of Earth's value.

• Example: Comparing the height reached by a ball thrown upward on Earth and on the Moon.

Work, Energy, and Power

Kinetic and Potential Energy

Energy is the ability to do work. Kinetic energy is due to motion, and potential energy is due to position.

• Kinetic Energy:

• Potential Energy (gravity):

• Work-Energy Theorem: The net work done on an object equals its change in kinetic energy.

Vectors and Vector Addition

Vector Operations

Vectors have both magnitude and direction. They can be added using the parallelogram rule or by components.

• Magnitude of a Vector:

• Example: Adding vectors and to find .

Applications and Problem Solving

Inclined Planes

When analyzing motion on an inclined plane, resolve forces parallel and perpendicular to the surface.

• Normal Force:

• Component of Gravity Down the Slope:

• Example: Calculating the acceleration of a child sliding down a hill with and without friction.

Elevator Problems

When an object is in an accelerating elevator, the normal force changes depending on the direction of acceleration.

• Apparent Weight: (upward acceleration), (downward acceleration)

Tables

Sample Table: Forces on Stacked Blocks

The following table summarizes the forces acting on stacked blocks as described in the exam questions:

Additional info: Values inferred using for each block and summing the masses above each block.

Key Concepts and Equations

• Newton's Second Law:

• Kinetic Friction:

• Centripetal Acceleration:

• Projectile Range:

• Kinetic Energy:

• Potential Energy:

• Vector Magnitude:

Additional Info

• Many questions require understanding of the effects of gravity on different planets (e.g., Moon vs. Earth).

• Some problems involve air resistance, which is often proportional to velocity or velocity squared, depending on the Reynolds number.

• For objects in free fall or on inclined planes, always resolve forces and apply Newton's laws appropriately.