One-Dimensional (1d) Motion

Terminology and Definitions

• Position: Specified by a single coordinate (e.g., x) on the axis of motion. Position can be positive or negative depending on the chosen reference point.

• Displacement: The change in position of an object. It can be positive or negative.

• Net Displacement: The sum of all individual displacements, or the difference between the final and initial positions.

• Distance: The absolute value of displacement for motion without turning points; always positive.

• Total Distance: The sum of all individual distances traveled; always positive.

• Warning: Total distance is not necessarily equal to the absolute value of net displacement.

Average vs. Instantaneous Quantities

• Velocity: The rate of change of position (can be positive or negative, always has the same sign as displacement).

• Average Velocity: Net displacement divided by total time interval.

• Speed: The absolute value of velocity (always positive, less informative than velocity).

• Average Speed: Total distance divided by total time interval.

• Uniform Linear Motion: Motion with constant velocity.

• Acceleration: The rate of change of velocity (can be positive or negative, depending on the choice of axes).

Mathematical Definitions:

• Instantaneous velocity:

• Instantaneous acceleration:

Note: Average quantities are not very useful for non-uniform motion.

1d Motion with Constant Acceleration

• Acceleration:

• Velocity:

• Position:

• Other Kinematic Equations for Displacement:

• Example: Free fall is a common example of motion with constant acceleration (assuming air resistance and elevation change are negligible).

Typical Plots for 1d Motion with Constant Acceleration

• Sign of acceleration (), initial velocity (), and initial position () depends on the details of the problem and choice of axes.

• Object speeds up if and have the same sign; slows down if they have opposite signs.

Two-Dimensional (2d) Motion

2d Motion with Constant Acceleration

• Acceleration: and (not necessarily equal).

• Velocity: ,

• Position: ,

• Both x and y components are needed to fully describe the motion.

• Free Fall: , (if upward is positive). is the acceleration due to gravity.

Projectile Motion

• Horizontal acceleration:

• Vertical acceleration:

• Horizontal velocity: (constant)

• Vertical velocity:

• Horizontal position:

• Vertical position:

• Trajectory: Path followed by projectile, described by (a downward parabola).

• Range: Total horizontal distance traveled.

• Peak: Highest point reached by the projectile.

Relative Motion

Reference Frames and Relative Velocity

• The trajectory and equations of motion depend on the choice of reference frame.

• Earth is typically used as the reference frame (assumed static).

• If the system is in motion with respect to a moving reference frame (MRF):

• Each velocity may be time-dependent.

2d Motion with Non-Constant Acceleration

• Acceleration: but is a known function of time.

• Velocity: ,

• Position: ,

• Previous kinematic equations are not valid for non-constant acceleration.

Forces and Motion

Newton's Laws and Types of Forces

• Force: A push or pull exerted by an object on another; measured in Newtons (N).

• Net Force: The vector sum of all forces acting on a system.

• Newton's Laws:

  1. 1st Law: An object remains at rest or in uniform motion unless acted on by a nonzero net force.

  2. 2nd Law: (acceleration is proportional to net force and inversely proportional to mass).

  3. 3rd Law: For every action, there is an equal and opposite reaction (forces act on different systems).

Types of Forces

• Gravitational Force: , points vertically downward.

• Normal Force: Exerted by the surface; always perpendicular to the contact surface.

• Tension Force: Exerted by a string/rope/cable; always along the string and same at both ends for an ideal string.

Problem-Solving Strategy for Forces and Motion

1. Identify the system of interest.

2. List all forces acting on the system and draw the free-body diagram (FBD).

3. Choose the best coordinate system (minimize projections).

4. Write Newton's 2nd law and project onto axes.

5. Solve for unknowns.

Friction and Other Forces

Kinetic and Static Friction

• Kinetic Friction: ; opposes relative motion, acts parallel to the interface.

• Static Friction: ; opposes the initiation of motion, acts parallel to the interface, .

Spring and Drag Forces

• Spring Force (Hooke's Law): ; always points toward equilibrium.

• Drag Force: For low speeds, ; for high speeds, ; always opposes motion.

• Terminal Velocity: The maximum velocity when drag force balances other forces (acceleration = 0).

Circular Motion

Uniform Circular Motion

• Object moves in a circle of radius at constant speed .

• Velocity is always tangent to the path; acceleration is centripetal (radially inward).

• Angular speed is constant.

Non-Uniform Circular Motion

• Speed is not constant; acceleration has both radial and tangential components.

• Time period:

• Frequency:

Work and Energy

Work

• Work is energy transferred by a force acting over a distance. For constant force and angle:

• Positive work: force in direction of displacement; negative work: force opposite to displacement; zero work: force perpendicular to displacement.

Kinetic Energy and Work-Energy Theorem

• Kinetic energy:

• Work-kinetic energy theorem:

• Work is a process quantity; kinetic energy is a state quantity.

Potential Energy and Conservation of Energy

Conservative Forces and Potential Energy

• Work done by conservative forces is path-independent; potential energy function can be defined.

• Change in potential energy:

• Gravitational:

• Elastic:

Energy Conservation

• Work-kinetic energy theorem: (where is work by non-conservative forces)

• Mechanical energy:

• Mechanical energy is conserved () if only conservative forces do work.

Force-Potential Energy Relationship

• (gradient operator gives force from potential energy)

• Potential energy curves: Minima correspond to stable equilibrium, maxima to unstable equilibrium.

• Power: [Watt]

Universal Gravitation

Newton's Law of Universal Gravitation

• For point masses (or spherically symmetric bodies):

• Properties: Always attractive, points along the line joining centers, follows inverse square law.

Gravitational Field

• Defined as force per unit test mass:

• Points radially inward, magnitude decreases with distance, units of acceleration.

Gravitational Potential Energy and Orbits

Potential and Mechanical Energy in Gravity

• Gravitational potential energy:

• Mechanical energy:

• Bound system: (closed orbit); unbound: (open orbit); escape speed: ,

Circular Orbits

• Orbital speed:

• Time period:

• Mechanical energy in orbit:

Linear Momentum and Collisions

Linear Momentum

• Momentum:

• Newton's 2nd law:

• For isolated systems: (momentum conserved)

• Momentum conservation is crucial for analyzing collisions and explosions (brief events, negligible external force).

Types of Collisions

• Totally Inelastic: Objects stick together; only momentum is conserved, not kinetic energy.

• Partially Inelastic: Objects deform but do not stick; only momentum is conserved, kinetic energy changes less than in totally inelastic events.

• Elastic: Objects do not stick or deform; both momentum and kinetic energy are conserved.