Projectile Motion and Newton's Laws

Projectile Motion

• General Solutions for Projectile Motion

  • Previously, specific problems were solved; now, the focus is on general solutions.

• Vectors in Two Dimensions

  • A vector can be split into horizontal (x) and vertical (y) components.

  • Horizontal component: $x \hat{i}$ where x is magnitude and $\hat{i}$ is the unit vector specifying direction.

  • Vertical component: $y \hat{j}$ where y is magnitude and $\hat{j}$ is the unit vector specifying direction.

  • Distinction between $v<em>x$ (scalar) and $\vec{v</em>x}$ (vector): Scalar multiplied by a vector ($\hat{i}$) yields a vector.

• Vector Addition

  • Algebraic Way: Add x-components and y-components separately, component by component.

  • Geometric Way: Tail-to-tip method. Place the tail of the second vector at the tip of the first vector; the resultant vector starts at the tail of the first and ends at the tip of the second.

• Equivalent Ways of Representing Vectors

  • Specify components.

  • Specify magnitude and angle.

  • Conversion formulas:

    • $\sin(\theta) = \frac{y}{\text{magnitude}}$

    • $\cos(\theta) = \frac{x}{\text{magnitude}}$

    • $\tan(\theta) = \frac{y}{x}$

    • $\text{magnitude} = \sqrt{x^2 + y^2}$

    • To find the angle : $\theta = \arctan(\frac{y}{x})$

• Projectile Motion as 2D Motion

  • Initial kick given in horizontal and vertical directions.

  • Horizontal direction: constant velocity motion (x doesn't change).

  • Vertical direction: influenced by gravity, acceleration g.

• Solving Projectile Motion Problems

  • Separate x and y coordinates.

  • Apply 1D motion equations to each axis.

  • Time (t) is the shared variable between both dimensions.

  • Acceleration in x is typically zero unless specified; in y, it's -g.

  • Two types:

    • Object shot from elevation with initial horizontal velocity (traces a parabolic shape).

    • Object shot from the ground at an angle (traces a full parabola).

• Characteristics of Projectile Motion from the Ground

  • At the peak, the vertical component of velocity (vy) is zero.

  • Horizontal component of velocity (vx) remains constant.

  • Vertical velocity points upwards before the peak and downwards after.

  • Maximum range is achieved when shooting at a 45-degree angle (hand-waving argument based on time in air and horizontal velocity).

• General Solution for Maximum Range

  • Solve problems with symbols (general names) instead of plugging in numbers immediately to reduce mistakes.

  • Assumptions: level ground, initial speed $\vec{v_0}$ at angle $\theta$.

• Derivation of Equations

  • Time to reach peak: $t<em>{peak} = \frac{v</em>{0y}}{g}$

  • Vertical component of initial velocity: $v<em>{0y} = v</em>0 \sin(\theta)$

  • Horizontal component of initial velocity: $v<em>{0x} = v</em>0 \cos(\theta)$

  • Time for the full trajectory: $t<em>{total} = 2 t</em>{peak} = \frac{2 v_0 \sin(\theta)}{g}$ (twice the time to reach the peak).

  • Horizontal distance traveled: $x = v<em>x \cdot t = v</em>0 \cos(\theta) \cdot \frac{2 v<em>0 \sin(\theta)}{g} = \frac{2 v</em>0^2 \sin(\theta) \cos(\theta)}{g}$

  • Using trigonometric identity: $2 \sin(\theta) \cos(\theta) = \sin(2\theta)$

  • General solution for projectile motion range: $x = \frac{v_0^2 \sin(2\theta)}{g}$

• Analysis of the Solution

  • To maximize range, maximize $\sin(2\theta)$, which occurs when $\sin(2\theta) = 1$ (90 degrees).

  • Therefore, $2\theta = 90^\circ$ which implies $\theta = 45^\circ$.

• Effects of Air Resistance

  • Air resistance slows down the motion, affecting the trajectory (shorter range).

  • The angle for maximum range will be altered.

• Relative Velocity

• Concept

  • Applies vector addition to relative motion.

  • Notation: $V_{PG}$, velocity of person (P) relative to ground (G).

• Rule:

  • If two indices are common when adding vectors (e.g., combining reference frames), they cancel out.

    • Example: $V<em>{PT} + V</em>{TG} = V_{PG}$

• Example

  • Person walking in a train: $V<em>{PG} = V</em>{PT} + V_{TG}$

    • Velocity of person relative to ground = velocity of person relative to train + velocity of train relative to ground.

• River Crossing Problem

• Problem

  • A boat crossing a river needs to account for the river's current.

  • To travel straight across, the boat must aim at an angle upstream.

• Solution Approach

  • $V_{BS} Velocity of boat (B) relative to shore (S)$

  • $V<em>{BS}= V</em>{BW} + V_{WS}$, velocity of the boat relative to the water + velocity of the water relative to the shore.

• Calculations

  • If $|V<em>{BW}| = 14$ and $|V</em>{WS}| = 3$ , then by Pythagorean theorem, $|V_{BS}| = \sqrt{14^2 - 3^2} = \sqrt{187} \approx 13.7$

  • Angle: $\theta = \arctan(\frac{3}{13.7}) \approx 12.4^\circ$. This angle must be aimed upstream.

• Important Note: Velocities are vectors, same mathematical principles apply as in the displacement vector cases. Any two vectors can be added in that fashion

Summarization

• Quantity with magnitude and direction is a vector, quantity with magnitude and no direction is a scalar.

• Projectile motion is an object near the earth surfaces under the influence of gravity.

• This is going to allow move to Newton Equation

Newton's Laws of Motion

• Historical Context

  • Isaac Newton's three laws: foundational for describing motion.

  • Applicable to large-scale objects at normal speeds.

• Limitations of Newton's Laws

  • Replaced by theories like relativity for extremely high speeds (near the speed of light).

• Relativity's Impact

  • Relativity introduces a factor of v/c (velocity over speed of light) modifying Newton's equations.

  • For everyday speeds (e.g., 60 mph), the relativistic effect is negligible (\approx 10^{-12}).

  • Newton's laws are a good approximation for everyday objects.

• Key Concepts

  • Force. Understand what is the force.

  • Inertia. What is inertia of and object

  • Newton's three laws describe all types of motion.

• Force Defined

  • Force is a push or pull.

  • A force changes an object's velocity (i.e., causes acceleration).

  • Measured using a scale.

  • Vector quantity (magnitude and direction).

• Inertia (Newton's First Law)

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

  • Equivalent terms: constant velocity, no acceleration, no net force.

  • Newton's First Law (Law of Inertia): An object continues in its state of rest or uniform velocity unless acted upon by a net force.

• Inertia and Mass

  • Inertia is directly proportional to mass.

  • Higher mass implies higher inertia (greater resistance to changes in motion).

• Newton's First Law Explained

  • An object maintains its inertia unless an external force acts upon it. This can be either object at rest in which case $\\vec v=0$, or object has some constant velocity $\\vec v \ne 0$

• Net Force

  • When multiple forces act on an object, the net force is the vector sum of all forces.

  • If the net force is zero, the object remains at rest (if initially at rest) or continues moving with constant velocity (if initially in motion).

• Inertial Reference Frames

  • Inertial reference frame: a frame moving at constant velocity relative to another.

  • Newton's First Law is valid in inertial frames.

  • Accelerating or rotating frames are non-inertial.

• Mass vs. Weight

  • Mass: Measure of inertia, measured in kilograms (kg).

  • Weight: Force experienced due to gravity.

  • Weight depends on gravitational acceleration (different on different planets).

  • In outer space, weight is zero due to lack of gravity, but mass remains unchanged.

• Newton's Second Law

• Equation:

  • $\vec{F_{net}} = m \vec{a}$ (Net force equals mass times acceleration).

• Interpretation

  • Force and acceleration, positive correlation.

  • Increased force increases acceleration if mass remains constant. Negative correlation between acceleration and mass.

  • The bigger mass value, acceleration will be bigger given the same force. Acceleration will be smaller given the same force.

• Relationship to First Law

  • If $\vec{F_{net}} = 0$, then $\vec{a} = 0$ (consistent with Newton's First Law).

  • If \vec{F_{net}} $\ne$ 0, then $\vec{a}$\ne 0$(acceleration occurs).</p></li></ul></li><li><p><strong>Force as a Vector</strong></p><ul><li><p>Forces can be split into components (like projectile motion).</p></li><li><p>Solve Newton's Second Law separately for x and y directions.</p></li></ul></li><li><p><strong>Units of Force</strong></p><ul><li><p>SI unit: Newton (N).</p></li><li><p>Newton is a composite unit (kg \cdot m/s²).</p></li></ul></li><li><p><strong>Weight calculation through math</strong></p></li><li><p>Calculating the weight</p><ul><li><p>$F_{weight} = mg$, where g is 9.8. The higher the g, the greater weight. Also the higher the mass, the greater the weight.</p></li></ul></li><li><p><strong>Newton's Third Law</strong></p></li><li><p>Newton's rule says that whenever one object exerts a force on the second object, the second object is also exerted equal force in the opposite direction to the first object</p></li><li><p><strong>Law</strong></p><ul><li><p>For every action, there is an equal and opposite reaction.</p></li><li><p>If object A exerts a force on object B, then object B exerts an equal and opposite force on object A.</p></li><li><p>Reaction and reaction are the same amount, but opposite directions.</p></li></ul></li><li><p><strong>Examples</strong></p><ul><li><p>Pushing a shopping cart: The cart also pushes back on you.</p><ul><li><p>You are standing on the ground, so the cart pushing you is canceled out by force to push yourself, but it can cause problems.</p></li></ul></li><li><p>Ice skating: If you push a wall, you push them other way, you feel go the other direction</p></li></ul></li><li><p><strong>Force felt by a Person versus a force from the air</strong></p><ul><li><p>Force from gravity creates an acceleration (g), and there is another force resisting that. Gravity is created because of earth and the reaction force is the force you apply to the ground and the force ground gives back</p></li></ul></li><li><p><strong>Convention:</strong></p><ul><li><p>$f_{12}$$ First, subscript is that the object of the force is not and the other is a source object