Fundamental Concepts in Introductory Physics: Kinematics, Dynamics, and Forces

Projectile Motion

• Maximum Range: The distance traveled by a projectile is maximized when the launch angle is 45° above the horizontal.

• Key Equation for Range:

$R = \frac{v_0^2 \sin 2\theta}{g}$

• Time of Flight:

$T = \frac{2v_0 \sin \theta}{g}$

• Maximum Height:

$H = \frac{v_0^2 \sin^2 \theta}{2g}$

• Effect of Angle: Launching at angles other than 45° reduces the range for a given initial speed.

• Comparing Planets: The range and height are affected by the acceleration due to gravity, which varies between planets (e.g., the Moon vs. Earth).

Relative Motion and Vectors

determining distances or velocities when moving in different directions. Vectors are used to represent quantities with both magnitude and direction.

• Vector Addition: The sum of two vectors is found using the parallelogram rule or by adding components.

• Perpendicular Vectors: If two vectors are perpendicular, the magnitude of their sum is given by:

$|\vec{A} + \vec{B}| = \sqrt{A^2 + B^2}$

• Traveling in Perpendicular Directions: If you travel 1 mile north, then 1 mile east, your distance from the starting point is:

$\sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.41$ miles

Dynamics: Forces and Newton's Laws

Newton's Laws of Motion

Newton's laws describe the relationship between forces and motion.

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

• Second Law: The net force on an object equals mass times acceleration:

$F = ma$

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

Friction

Friction is a force that opposes motion between two surfaces in contact.

• Static Friction: The force that must be overcome to start moving an object. Maximum static friction is:

$f_{s,\text{max}} = \mu_s N$

• Kinetic Friction: The force opposing motion once an object is sliding:

$f_k = \mu_k N$

• Inclined Planes: The angle at which an object starts to slide is determined by the coefficient of static friction:

$\tan \theta = \mu_s$

• Applications: Friction is crucial in problems involving cars on curves, objects on ramps, and sleds on slopes.

Normal Force

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

• On a Flat Surface: $N = mg$

• On an Inclined Plane: $N = mg \cos \theta$

Tension

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

• Rope in Tug-of-War: The tension equals the force applied by each team if the rope is in equilibrium.

• Multiple Masses and Strings: The maximum tension a string can withstand determines which string breaks in a system of connected masses.

Circular Motion

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 Acceleration:

$a_c = \frac{v^2}{r}$

• Centripetal Force:

$F_c = m \frac{v^2}{r}$

• Applications: Cars turning on curves, objects on circular tracks, and satellites in orbit.

Work, Energy, and Power

Kinetic and Potential Energy

Energy is the capacity to do work. Kinetic energy is energy of motion, and potential energy is stored energy due to position.

• Kinetic Energy:

$KE = \frac{1}{2}mv^2$

• Potential Energy (Gravity):

$PE = mgh$

• Conservation of Energy: In the absence of non-conservative forces (like friction), total mechanical energy is conserved.

Gravity and Free Fall

Acceleration Due to Gravity

Objects in free fall near the Earth's surface accelerate downward at $g \approx 9.8\, \text{m/s}^2$ (on the Moon, $g$ is much less).

• Free Fall Equations:

$y = v_0 t + \frac{1}{2} a t^2$

• Time to Fall from Height $h$:

$t = \sqrt{\frac{2h}{g}}$

Forces in Systems of Masses

Stacked Masses and Normal Forces

When multiple masses are stacked, the normal force on each mass depends on the weight of the masses above it.

Mass	Normal Force from Table
Bottom mass	Sum of all masses × $g$
Middle mass	Sum of masses above × $g$
Top mass	Its own mass × $g$

Air Resistance and Terminal Velocity

Terminal Velocity

Terminal velocity is the constant speed reached by an object when the force of air resistance equals the force of gravity.

• Forces at Terminal Velocity:

$mg = F_{\text{drag}}$

• Drag Force (for high Reynolds number):

$F_{\text{drag}} = C v^2$

• Terminal Velocity Equation:

$v_t = \sqrt{\frac{mg}{C}}$

• Comparing Spheres: Terminal velocity depends on radius and density; for spheres of different sizes and densities, the ratio can be found using the above formula.

Sample Applications and Problem Types

• Inclined Planes: Calculating the angle at which objects begin to slide using friction coefficients.

• Elevator Problems: Determining apparent weight when accelerating upward or downward.

• Collisions: Conservation of momentum applies when two objects collide in mid-air.

• Ropes and Tension: Analyzing forces in tug-of-war or hanging masses to find tension and breaking points.

• Vector Problems: Adding and comparing magnitudes of vectors in different directions.

Additional info:

• Some questions require knowledge of trigonometry (e.g., resolving forces on ramps).

• Air resistance is often modeled as proportional to $v^2$ for large, fast-moving objects (high Reynolds number).

• For circular motion, acceleration and velocity vectors are perpendicular.