Kinematics and Dynamics in Classical Mechanics

Kinematic Equations

Kinematics describes the motion of objects without considering the causes of motion. The following equations are fundamental for analyzing motion with constant acceleration:

• Displacement (with constant acceleration):

• Final velocity (from initial velocity and acceleration):

• Final velocity squared (from displacement):

Key Terms:

• : Displacement (meters, m)

• : Velocity (meters per second, m/s)

• : Acceleration (meters per second squared, m/s2)

• : Time (seconds, s)

Example: Calculating the stopping distance of a train using initial velocity, acceleration, and kinematic equations.

Newton's Second Law

Newton's Second Law relates the net force acting on an object to its mass and acceleration:

• Equation:

• Key Terms:

  • : Net force (Newtons, N)

  • : Mass (kilograms, kg)

  • : Acceleration (m/s2)

Application: Used to analyze forces in problems involving motion, such as sleds being pulled or objects sliding down ramps.

Weight

Weight is the force of gravity acting on an object:

• Equation:

• Key Terms:

  • : Weight (N)

  • : Mass (kg)

  • : Acceleration due to gravity ( on Earth)

Example: Calculating the normal force on a sled or the force required to pull an object at an angle.

Friction

Friction is a resistive force that opposes the motion of objects in contact. There are two main types:

• Static friction (): Prevents motion up to a maximum value.

• Kinetic friction (): Opposes motion once sliding begins.

• Normal force (): The perpendicular contact force exerted by a surface.

Key Terms:

• : Coefficient of static friction

• : Coefficient of kinetic friction

• : Normal force (N)

Example: Determining the coefficient of kinetic friction for a train stopping or a sled moving at constant speed.

Drag Force

Drag is a resistive force experienced by objects moving through a fluid (such as air):

• Equation:

• Key Terms:

  • : Drag force (N)

  • : Drag coefficient (dimensionless)

  • : Fluid density (kg/m3)

  • : Cross-sectional area (m2)

  • : Velocity (m/s)

Application: Important in analyzing motion at high speeds or in fluids, such as cars, airplanes, or falling objects.

Applications and Example Problems

Problem 1: Locomotive Stopping Distance and Friction

Scenario: A 720,000-kg locomotive traveling at 120 km/h must stop after the brakes are applied, decelerating at 2.65 m/s2.

• (a) Stopping Distance: Use kinematic equations to find the distance required to stop.

• (b) Coefficient of Kinetic Friction: Use Newton's laws and friction equations to determine .

Answers:

• a. 209 m

• b. 0.270

Problem 2: Projectile Motion – Arrow Shot Horizontally

Scenario: An arrow is shot horizontally from a height of 1.5 m at 12 m/s.

• (a) Time to Hit Ground: Use vertical motion equations to find time.

• (b) Vertical Speed at Impact: Use for vertical component.

Answers:

• a. 0.55 s

• b. -5.4 m/s

Problem 3: Sled Pulled at an Angle with Friction

Scenario: A 22-kg sled is pulled at 32° above the horizontal on grass () at constant speed.

• (a) Free-Body Diagram: Draw and label all forces (tension, gravity, normal, friction).

• (b) Normal Force: Calculate the normal force using force components and Newton's laws.

Answer: 183 N

Problem 4: Hockey Puck Sliding Down an Inclined Ramp

Scenario: A 200 g puck slides down a 40° ramp (length 1.8 m, , ).

• (a) Free-Body Diagram: Draw and label all forces (gravity, normal, friction).

• (b) Final Speed: Use energy or kinematic analysis to find speed at the bottom.

Answer: 3.8 m/s

Summary Table: Key Equations and Their Applications

Additional info: These notes are based on a typical introductory physics exam covering kinematics, Newton's laws, friction, and applications to real-world problems. Free-body diagrams are essential for force analysis in all problems involving Newton's laws.