Kinematics and Newtonian Mechanics: Study Guide

Kinematics

Displacement and Distance

Displacement is a vector quantity that refers to the change in position of an object. It is different from distance, which is a scalar and only measures the total path length traveled.

• Displacement: The straight-line distance from the initial to the final position, with direction.

• Distance: The total length of the path traveled, regardless of direction.

• Formula for displacement (in 1D): $\Delta x = x_f - x_i$

• Example: If a postal worker walks 161 m east, then 194 m due west, the displacement is $161\,\text{m} - 194\,\text{m} = -33\,\text{m}$ (33 m west).

Average Speed and Velocity

Average speed is the total distance divided by the total time, while average velocity is the total displacement divided by the total time.

• Average speed: $\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}$

• Average velocity: $\text{Average velocity} = \frac{\text{Displacement}}{\text{Total time}}$

• Example: If a car travels at different speeds for different time intervals, the average speed for the entire trip can be found by dividing the total distance by the total time.

Acceleration

Acceleration is the rate of change of velocity. It is a vector quantity and can be caused by changes in speed or direction.

• Formula: $a = \frac{\Delta v}{\Delta t}$

• Westward acceleration: Occurs when an object increases its velocity in the westward direction or decreases its eastward velocity.

• Example: A car traveling eastward and slowing down has a westward acceleration.

Projectile Motion

Projectile motion involves two-dimensional motion under the influence of gravity, with horizontal and vertical components analyzed separately.

• Horizontal motion: Constant velocity (if air resistance is neglected).

• Vertical motion: Constant acceleration due to gravity ($g = 9.8\,\text{m/s}^2$ downward).

• Key equations:

  • Horizontal displacement: $x = v_{0x} t$

  • Vertical displacement: $y = v_{0y} t - \frac{1}{2} g t^2$

  • Time of flight (for vertical drop): $y = \frac{1}{2} g t^2 \implies t = \sqrt{\frac{2y}{g}}$

  • Range of projectile: $R = \frac{v_0^2 \sin 2\theta}{g}$

• Example: A rock kicked horizontally from a cliff: use $x = v_{0x} t$ and $y = \frac{1}{2} g t^2$ to solve for time and range.

Relative Velocity

Relative velocity is the velocity of an object as observed from a particular reference frame.

• Vector addition: $\vec{v}_{A/B} = \vec{v}_A - \vec{v}_B$

• Example: A car turns from east to south at constant speed; the change in velocity is a vector difference between the two directions.

Newton's Laws and Forces

Newton's First Law (Inertia)

An object at rest remains at rest, and an object in motion remains in motion at constant velocity unless acted upon by a net external force.

• Inertia: The tendency of an object to resist changes in its state of motion.

Newton's Second Law

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

• Formula: $\vec{F}_{\text{net}} = m \vec{a}$

• Example: If a net force $F$ is applied to mass $m$, $a = \frac{F}{m}$.

• Doubling mass: If mass is doubled, acceleration is halved for the same force.

Weight and Normal Force

Weight is the force of gravity acting on an object. The normal force is the perpendicular contact force exerted by a surface.

• Weight: $W = m g$

• Normal force: On a flat surface, $N = m g$; on an incline, $N = m g \cos \theta$

• Example: The weight of a 2.50-kg bag on Earth: $W = 2.50 \times 9.8 = 24.5\,\text{N}$

Friction

Friction is the resistive force that opposes the relative motion of two surfaces in contact.

• Static friction: Prevents motion up to a maximum value $f_s \leq \mu_s N$

• Kinetic friction: Opposes motion, $f_k = \mu_k N$

• Example: A block against a wall may remain at rest if static friction is sufficient to balance the applied force.

Tension

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

• Inclined plane: Tension in a rope holding a mass $M$ on an incline at angle $\theta$ is $T = M g \sin \theta$ (if the rope is parallel to the incline and neglecting friction).

• Example: For a 10-kg block on a 30° incline, $T = 10 \times 9.8 \times \sin 30^\circ = 49\,\text{N}$

Force Components

Forces can be resolved into components along the x and y axes using trigonometric functions.

• Horizontal component: $F_x = F \cos \theta$

• Vertical component: $F_y = F \sin \theta$

• Example: Two forces act on a block at angles; sum the x-components to find net horizontal force and use $F_{\text{net}} = m a$ to find acceleration.

Sample Table: Comparison of Friction Types

Type of Friction	Symbol	Formula	When it Applies
Static Friction	$f_s$	$f_s \leq \mu_s N$	Object at rest
Kinetic Friction	$f_k$	$f_k = \mu_k N$	Object in motion

Applications and Problem-Solving Strategies

• Draw a free-body diagram to identify all forces acting on an object.

• Resolve forces into components along chosen axes (usually horizontal and vertical).

• Apply Newton's laws to set up equations for unknowns.

• Use kinematic equations for motion problems, especially when acceleration is constant.

• Check units and directions for consistency in calculations.

Key Equations Summary

• $v = v_0 + a t$

• $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• $v^2 = v_0^2 + 2 a (x - x_0)$

• $F = m a$

• $W = m g$

• $f_s \leq \mu_s N$

• $f_k = \mu_k N$

• $T = M g \sin \theta$ (for tension on an incline)

Example Application: A 25 N net force is applied to a 12-kg box for 5.7 s from rest. The acceleration is $a = \frac{25}{12} = 2.08\,\text{m/s}^2$. The final speed is $v = 0 + (2.08)(5.7) = 11.86\,\text{m/s}$.

Additional info: This study guide covers the main concepts and problem types found in introductory college physics, focusing on kinematics, Newton's laws, forces, friction, and tension, as reflected in the provided questions.