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GenPhysics Notes 1st Quarter AHow mLesson 1: Physical...

GenPhysics Notes 1st Quarter AHow mLesson 1: Physical English units (aka Imperial units) - Quantities unlike the SI units, does not have a Physical Quantities - is the property of a rule for its physical quantities as a material to be quantified through measurement whole but each unit has a conversion (E.g. length, width, time, etc.). Where it is expressed in terms of units of measurement. Units of Measurement: Units of measurement can be classified as Base units and Derived units where the base units only measures a single physical quantity of an object/material wherein derived units quantified 2 or more objects (E.g. 2 is considered derived as it quantified both length and time) ratio/factor. E.g 1 feet = 12 inches Figure 3: Imperial Unit Conversion Table Conversion of Units: To convert units, the conversion factor for the 2 units are considered. Metric-to-Metric: uses the base 10 for units (refer to figure 2) Imperial-to-Imperial: each unit has a unique conversion factor with regards to their specific physical quantity (refer Figure 1: SI base units to figure 3) Metric-to-Imperial: uses a conversion SI units (Systéme International units) factor for each physical quantity (refer - uses a base 10 for units and are to figure 4) defined by the prefixes of the base word of the physical quantity being English Metric described. E.g KILOmeter, 1 in 2.54 cm 1 mi 1.61 km 1 oz 28 g 1 pound 450 g 1 gallon 4L Figure 4: Metric-to-Imperial Conversion Table Scientific Notation - a way of expressing number to show extreme numbers in a shortened form Rules for counting significant figures: Non-zero digits and zeros in between nonzero digits are considered significant. Leading zeros before the decimal CENTImeter. point does NOT count. Figure 2: SI unit conversion table Trailing zeros count with the presence of a decimal point. Trailing zeros may or may not count without the presence of a decimal The document does not in any way claim property of the following content but serves as a material for only education purposes. Follow @AkhikoM on IG.  GenPhysics Notes 1st Quarter point thus the number of significant figures is unknown. Lesson 2: Errors and Uncertainty Variance and Standard Deviation: Accuracy and Precision: Variance is the variability of data with regards Accuracy - closeness of data to its to the expected value or mean. Standard true value. Deviation is the square root of the variance. Precision - closeness of data to one another. Lesson 3: Scalar and Vector High accuracy may mean high precision but Quantities high precision cannot be translated to high Scalar quantities - physical quantities that accuracy. possess only magnitude (e.g. speed) Vector quantities - physical quantities that Errors and Uncertainty: possess both magnitude and direction (e.g. Error: is defined as the difference velocity) between the target value and the observed/measured value. Parts of a vector: Uncertainty: describes the reliability A vector is usually represented by an of the assertion that the stated arrow in which the arrow’s end (tail) point is measurement result represents the the initial position, and the tip (head) being the value of the measurand. final position. Wherein the length of the arrow shows magnitude. Kinds of error: Random Error - are errors from unpredictable or inevitable changes during data measurement. Often linked with accuracy. Systematic Error - usually comes from the measuring instrument or in the design of the experiment itself which results in errors being observed Addition and Subtraction of Vectors to be consistent. Linked with precision. (graphically): The addition of vectors involves Estimation of Errors: connecting all vectors from one's head to Absolute error - error defined as the another's tail in which order does not matter difference between the measured (as long as the origin/starting point remains value to the actual/target value. the same/constant and the direction and Absolute Error = 𝑉𝐴 − 𝑉𝐸 magnitude of the vectors remains the same). Relative error - error defined as the ratio of the difference of the measured value and the actual/target value to the actual/target value. Simply the ratio between the absolute error and the actual value. 𝑉𝐴− 𝑉𝐸 Relative Error = 𝑉𝐸 The document does not in any way claim property of the following content but serves as a material for only education purposes. Follow @AkhikoM on IG.  GenPhysics Notes 1st Quarter The line connecting the endpoints is the sum 𝐶 = 𝐶𝑥 + 𝐶𝑦; 𝐶𝑥 = 𝐴𝑥 + 𝐵𝑥; 𝑎𝑛𝑑 𝐶𝑦 = 𝐴𝑦 + 𝐵𝑦 called the resultant. The subtraction of vectors functions the same as addition, however it involves turning the direction of the subtrahend to the opposite direction (180 degrees) from the original direction. Vectors as components: Vectors can be broken down into Pythagorean theorem: components such as x and y which represent their movement along the axes and is/can be represented as 𝑉 = 𝑉𝑥 + 𝑉𝑦 (Unit Vector Form). Using components, a right triangle will be formed using the axes and an angle will be formed from the x-axis. Having a right triangle invokes the usage of pythagorean theorem where the reference angle is from the x-axis. As such, trigonometric functions can be utilised when finding a specific vector using For getting the resultant/sum of these the axes. components we simply use the pythagorean 2 2 theorem. 𝑉 = 𝑉𝑥 + 𝑉𝑦 When it comes to multiple vectors we simply do the same thing however we add all the components among the axis first such that Using these functions, the following trigonometric equivalent can be obtained: x = R cos θ (in this case; b is x) y = R sin θ (in this case; a is y) −1 𝑦 θ = 𝑡𝑎𝑛 | 𝑥 | Unit Vectors: Unit vectors, by definition, are vectors components that are equivalent to positive 1 that represents direction in which is denoted The document does not in any way claim property of the following content but serves as a material for only education purposes. Follow @AkhikoM on IG.  GenPhysics Notes 1st Quarter by î + ĵ + k̂. These unit vectors are associated with the 3d plane system in which î represents Average Acceleration - refers to the change the x-value, ĵ for the y-value, and k̂ for the in speed/velocity over time. z-value. ∆𝑣 𝑣𝑓−𝑣𝑖 𝑎= ∆𝑡 = 𝑡𝑓−𝑡𝑖 Instantaneous vs Average Motion Average motion - the change in motion over a duration of time. Instantaneous motion - the change in motion in a specific amount of time. It is defined as the slope within a graph. Addition when it comes to unit vectors Instantaneous Velocity - the specific velocity involves adding each individual component. within a period of time. Is defined as the slope of a position vs time graph.or the derivative of the position equation. 𝑑𝑥 𝑣𝑖 = 𝑑𝑡 Instantaneous Acceleration - the specific Similarly, subtraction works the same acceleration within a period of time. Is defined way, but we negate the signs of the as the slope of a velocity vs time graph or the subtrahend then we add the vectors. derivative of the velocity equation. 𝑑𝑣 𝑎𝑖 = Lesson 4: Motion and Kinematics 𝑑𝑡 Mechanics - the branch of applied Note: mathematics dealing with motion and forces The reverse process can also be done producing motion. wherein the anti-derivative of acceleration = velocity, and the anti-derivative of velocity = Dynamics - a branch of mechanics in which position. involves the study of the cause of motion, or more precisely the cause of changes in motion. This can be divided into 2 further Lesson 5: Uniformly Accelerated branches. Motion (UAM) Kinematics - a branch of dynamics Uniformly Accelerated Motion that describes the motion of objects are motions in which the acceleration without reference to the forces which is constant/uniform cause the motion. Can be classified as either horizontal Kinetics - branch of dynamics that or free fall concerns the effect of forces and torques on the motion of bodies Horizontal Uniformly Accelerated Motion having mass. - Are UAMs that occurs along the x-axis/happens horizontally Position - refers to the location of an object within a specified frame of reference HUAM Equations Missing commonly denoted as x. ∆𝑥 = 𝑥𝑓 − 𝑥𝑖 x 𝑉𝑓 = 𝑉0 + 𝑎𝑡 Average Velocity - refers to the change in 1 2 ∆𝑥 = 𝑉0𝑡 + 𝑎𝑡 𝑉𝑓 position over time. 2 ∆𝑥 𝑥𝑓−𝑥𝑖 𝑉= ∆𝑡 = 𝑡𝑓−𝑡𝑖 The document does not in any way claim property of the following content but serves as a material for only education purposes. Follow @AkhikoM on IG.  GenPhysics Notes 1st Quarter 2 𝑉𝑓−𝑉0 2 t ∆𝑦 = 𝑉0𝑡 + 1 2 𝑔𝑡 𝑉𝑓 ∆𝑥 = 2𝑎 2 𝑉𝑓 +𝑉0 a 2 𝑉𝑓−𝑉0 2 t ∆𝑥 = ( )t ∆𝑦 = 2𝑔 2 𝑉𝑓 +𝑉0 g Free Fall Motion ∆𝑦 = ( 2 )t - state of a body that moves freely in any manner in the presence of gravity. - g is a constant that represents 2 acceleration due to gravity (9.8𝑚/𝑠 ) Obtaining the initial velocity of components Free Fall Equation Missing given the combined velocity and the angle: 𝑉𝑓 = 𝑉0 + 𝑔𝑡 y Note: Displacement of 2 different angles are the same given that their angles and 1 2 𝑉𝑓 conditions are the same. ∆𝑦 = 𝑉0𝑡 + 2 𝑔𝑡 Combined velocity is simply the addition 2 𝑉𝑓−𝑉0 2 t ∆𝑦 = between the velocity of the components on 2𝑔 both x and y-axes thus trigonometric 𝑉𝑓 +𝑉0 g equivalence is used to obtain individual ∆𝑦 = ( 2 )t velocities. Lesson 6: Projectile Motion Projectile Motion - is a special type of free fall motion that involves movement in both the x and y axis simultaneously. Movement along the axes: Note: the time it takes to reach the peak of the X-axis: The movement along the parabola and the time it falls after the peak is x-axis is similar to that of a horizontal the same. uniformly accelerated motion without 2 the acceleration (a = 0 m/𝑠 ). Additional Notes: range / the change in x can Y-axis: The movement along the also be computed using the following formula: y-axis is the same that of free fall. 2 𝑉0 𝑠𝑖𝑛 2θ ∆𝑥 = 𝑔 Equations regarding projectile motion: Movement along the x-axis Missing Lesson 7: Circular Motion Circular Motion - described as motions ∆𝑥 = 𝑉0𝑡 - following a circular path which can either be uniform or non-uniform. Note: time could not be obtained from the x Uniform vs Non-Uniform Circular Motion variable unless both the displacement and velocity is given. Uniform Non-Uniform Movement along the y-axis Missing Speed Constant Varies Velocity Changes Direction Varies Speed and 𝑉𝑓 = 𝑉0 + 𝑔𝑡 y Direction The document does not in any way claim property of the following content but serves as a material for only education purposes. Follow @AkhikoM on IG.  GenPhysics Notes 1st Quarter Acceleratio Constant Magnitude Varying Magnitude Properties of Non-Uniform Circular Motion n Direction and speed varies Does not have Has components components Magnitude of acceleration is not Perpendicular to Resultant between the constant velocity vector components x and y Has components of acceleration radial radial + tangential Centripetal acceleration: 𝑑|𝑣| 2 2 Angle Always Less than 90 between 𝑎= 𝑑𝑡 = 𝑎𝑡𝑎𝑛 + 𝑎𝑅 = 𝑎𝑡𝑎𝑛 + 𝑎𝑅 velocity and perpendicular degrees or less acceleration (90 degrees) that ½ π Note: Tangential acceleration is the component of vector acceleration tangent to the path while radial Summary of Difference acceleration is the component of acceleration point towards the center of the path. Uniform Circular Motion - Circular motion Lesson 8: Relative Motion that has acceleration due to varying direction Relative Motion - are motions observed from of velocity but has constant speed. a frame of reference. All motions are relative There is no absolute state of motion as it changes relative to the reference. Frame of reference - Vantage point with respect to position and motion it Coordinate system + time Notation: 𝑉𝑎𝑏; where a is the position/subject and b is the reference Properties of Uniform Circular Motion Has no components of acceleration Speed is always tangent to the path Acceleration is perpendicular to the path/velocity and is directed inwards Magnitude of acceleration is constant General Equations: Radial (centripetal) acceleration: 2 2 1. (1 dimensional) 𝑋𝑃𝐴 = 𝑋𝑃𝐵 + 𝑋𝐵𝐴 𝑎𝑟𝑎𝑑 = 𝑉 𝑅 = 4π 𝑟 2 ; (v is speed) 2. (2 dimensional) 𝑅𝑃𝐴 = 𝑅𝑃𝐵 + 𝑅𝐵𝐴 𝑡 𝑉= 2π𝑟 𝑡 ; (t is called period) 2 𝐹𝑐 = 𝑚𝑉 ; (centripetal force) Personal Notes: find the sum of relative 𝑟 motion is similar to adding line segments such that: Non-Uniform Circular Motion - Circular motion that has acceleration due to varying direction and speed. Lesson 9: Laws of Motion Laws of motion - Newton’s laws of motion, three statements describing the relations between the forces acting on a body and the motion of the body, first formulated by English physicist and mathematician Isaac Newton, The document does not in any way claim property of the following content but serves as a material for only education purposes. Follow @AkhikoM on IG.  GenPhysics Notes 1st Quarter which are the foundation of classical mechanics. First law of motion (law of inertia): “An object at rest remains at rest, and an object in motion remains in motion at constant speed and in a straight line unless acted on by an unbalanced force.” - This implies that all forces applied to an object are in an equilibrium ( Similar to vector addition, the resultant force is Σ𝐹 = 0) or the sum of all forces computed by getting the sum of the forces in equates to 0, unless acted upon by an the x-axis and the y-axis using the equation: external force. 𝑅 = 𝑅𝑥 2 2 + 𝑅𝑦 ; 𝑅𝑥 = Σ𝐹𝑥, 𝑅𝑦 = Σ𝐹𝑦 Second Law of Motion (law of Forces) - “The acceleration of an object depends on the mass of the object and the amount of force Free body diagram - is a sketch/ diagram of applied.” an object of interest with all the surrounding - Simply states that the magnitude of objects stripped away and all of the forces force applied is the magnitude of acting on the body shown. acceleration multiplied to the mass of - The object in focus is represented as a the object. period (.). - The forces are expressed in vector Force (units: Newtons) - is a push or a pull forms where the length of the arrows that changes or tends to change the state of signifies their magnitude. rest or uniform motion of an object or changes the direction or shape of an object. It causes objects to accelerate. Hence the equation: Force = acceleration * mass (F = am). Types of Forces: 1. Contact Forces - are forces which can only be applied when objects collide/ make contact with one another. 2. Non-contact Forces - are forces in which one does not need physical contact with one another. Contact Force Non-contact Force Getting the resultant of a free body diagram Normal Force Gravitational Force - To get the resultant of a free body Applied Force Magnetic Force diagram, the principle of superposition Frictional Force Long-ranged Forces in motion will be used in order to get Tension Force Electrical Force the resulting vector of all forces. Trigonometric functions will also be Principle of Superposition in motion - When utilized for forces that is not parallel to several charges interact, the total force on a the axis of the forces such that particular charge is the vector sum (or the net force) of the forces exerted on it by all other charges. The document does not in any way claim property of the following content but serves as a material for only education purposes. Follow @AkhikoM on IG.  GenPhysics Notes 1st Quarter Angle Formula Θ = 0 |a| * |b| Θ = 90 0 Θ = 180 - ( |a| * |b| ) - 0 < θ < 90 |a| * |b| * Cos Θ Third law of motion (law of action and 90 < θ < 180 - ( |a| * |b| * Cos Θ) reaction) - “Whenever one object exerts a - Using components, scalar product can force on another object, the second object be computed using the following exerts an equal and opposite force on the formula: first.” A * B = AxBx + AyBy + AzBz - Simply states that for every action there is an equal and opposite Properties of Scalar Products: reaction. 𝐹𝑎/𝑏 = − 𝐹𝑏/𝑎 U*V=V*U U * (V + W) = UV + UW C * (U * V) = CU * V = CV * U 0*V=0 2 V * V = |𝑉| Lesson 10: Scalar and Vector Application of Dot Product: Multiplication Work - is the energy transferred to or from an Types of Vector Multiplications: object via the application of force (F) along a 1. Scalar Multiplication - The displacement (s). Work can be computed by: multiplication between a scalar and vector quantity. W = Fs (when constant force) 2. Scalar Product / Dot Product - The W = ∫F * dx (when non-constant force) multiplication of a Vector to another vector to produce a scalar quantity. Additional Lesson 1: Cross Product 3. Cross Product / Vector Product - Cross Product - is a binary operation on two The multiplication of a vector to vectors in three-dimensional space. It results another vector to produce a vector in a vector that is perpendicular to both quantity. vectors. The Vector product of two vectors, a and b, is denoted by a × b. (Vector x Vector = Scalar Multiplication: Vector) - Simply works by multiplying a scalar quantity to a vector (Scalar x Vector) Angle Formula Θ = 0 0 Θ = 90 |a| * |b| Personal notes: scalar does not have Θ = 180 0 components thus you simply multiply their magnitudes 0 < θ < 90 |a| * |b| * Sin Θ 90 < θ < 180 - (|a| * |b| * Sin Θ ) Scalar Product / Dot Product: - obtained by multiplying the - Note: Maximum can be obtained when magnitudes of the vectors and the cos all vectors are orthogonal or angle between them. (Vector x Vector perpendicular. = Scalar) Cross Product using Components: The document does not in any way claim property of the following content but serves as a material for only education purposes. Follow @AkhikoM on IG.  GenPhysics Notes 1st Quarter - To compute the cross products given Lesson 12: Center of Mass and the components, first we must list the Center of Gravity components as such: Center of Mass: - a position defined relative to an object or system of objects. It is the unique position at which the weighted position vectors of all the parts of a system sum up to zero. - We then cross multiply the components outside the column of the unit of interest and get their difference. Formula for center of mass: Ξ𝑚𝑥 𝐶𝑀 = Ξ𝑚 CM - Center of Mass, m is the mass of the object, and x is the position of the object relative to the point. Example: Imagine a seesaw and the middle being the relative position: Lesson 11: Universal Law of Gravitation Universal Law of Gravitation - states that all Using the formula for center of mass we could objects attract each other with a force of find that the center of mass is 0m from the gravitational attraction. Gravity is universal. relative position. This force of gravitational attraction is directly dependent upon the masses of both objects Center of Gravity: and inversely proportional to the square of the - is the point through which the force of distance that separates their centers. Newton's gravity acts on an object or system conclusion about the magnitude of gravitational forces is summarized Difference between center of gravity and symbolically as mass: Center of mass is the point at which the 𝑚1*𝑚2 𝐹𝑔𝑎𝑣 ∝ distribution of mass is equal in all directions 2 𝑑 and does not depend on the gravitational field. Center of gravity is the point at which the distribution of weight is equal in all directions and it does depend on the gravitational field. The center of mass and center of gravity are in the same position if the gravitational field in Another means of representing the which the object exists stays uniform. proportionalities is to express the relationships in the form of an equation using a constant of Lesson 13: Potential Energy proportionality. Conservative Force - any force, such as the gravitational force between Earth and another 𝐺 * 𝑚1*𝑚2 mass, whose work is determined only by the 𝐹𝑔𝑎𝑣 = 2 𝑑 final displacement of the object acted upon. Where G represents the universal gravitational Examples: Gravitational and Elastic Forces −11 2 2 constant (𝐺 = 6. 673 𝑥 10 𝑁𝑚 /𝑘𝑔 ) Properties of Conservative Force: The document does not in any way claim property of the following content but serves as a material for only education purposes. Follow @AkhikoM on IG.  GenPhysics Notes 1st Quarter 1. The total work done by a conservative KE = ½ m𝑣 2 force is independent of the path Work Energy Theorem: 𝑊 = ∆𝐾𝐸 resulting in a given displacement and is equal to zero when the path is a closed loop. Potential Energy (represented by U) - The energy possessed by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors. Associated with the configuration (arrangement) of a system of objects that exerts forces on one another. ∆𝑈 = − 𝑊 Law of Conservation: 𝐸 = 𝐾𝐸 + 𝑈 𝑊1 = 𝑊2 = 𝑊3 2. work done by a conservative force or against a conservative force along a closed loop is zero. 𝑊𝐴𝐵 + 𝑊𝐵𝐴 = 0 3. Reversible from KE to PE, vice versa Nonconservative force - are dissipative forces such as friction or air resistance. These forces take energy away from the system as the system progresses, energy that you can’t get back. These forces are path dependent; therefore it matters where the object starts and stops. Examples: Friction, Air Resistance, Tension, Normal Force Kinetic Energy (represented by KE) - Form of energy that an object or a particle has by reason of its motion. If work, which transfers energy, is done on an object by applying a net force, the object speeds up and thereby gains kinetic energy. The document does not in any way claim property of the following content but serves as a material for only education purposes. Follow @AkhikoM on IG.  GenPhysics Notes 1st Quarter Gravitational Potential Energy - The energy Lesson 14: Impulse and Momentum an object possesses because of its position in Momentum a gravitational field. Associated with the state - a vector quantity who is a product of of separation between two objects that attract the mass of a particle and its velocity. each other by the gravitational force. The It is also known as inertia in motion. higher the distance apart, the higher the - Formula: P = mv potential energy. - The rate of change of momentum of U = mgh (m - mass, g - acceleration due to gravity, h - height) an object is proportional to is equal to the net force applied to it Elastic Potential Energy - The energy stored - Ξ𝐹 = ∆𝑃 ∆𝑡 as a result of applying a force to deform an elastic object. The energy is stored until the Impulse - Momentum Theorem force is removed and the object springs back - states that the impulse applied to an to its original shape, doing work in the object is equal to the change in its process. 2 momentum. U = ½k ∆𝑥 (k - spring constant, ∆𝑥 - displacement) Equilibrium - If an object is said to be in a state of equilibrium, then all the forces which act upon the object are balanced. It is defined as the state of rest due to the equal action of the opposing forces. Example: Equilibrium Points 1. Stable equilibrium: When a particle is displaced slightly from a position, then a force acting on it brings it back to the initial position, it is said to be in stable equilibrium position. 2. Unstable equilibrium: When a particle is displaced slightly from a position, then a force acting on it tries Impulse to displace the particle further away - Is the change in momentum from the equilibrium position, it is said - J = ∆𝑃 to be in unstable equilibrium. 3. Neutral Equilibrium: When a particle is slightly displaced from a position Formulas Derived from Momentum and then it does not experience any force Impulse acting on it and continues to be in 𝐹𝑛𝑒𝑡 equilibrium in the displaced position, it 𝑚 = 𝑎 is said to be in neutral equilibrium. Δ𝑣 𝑎= 𝑡 𝐹𝑛𝑒𝑡 Δ𝑣 𝑚 = 𝑡 Ft = Δmv Collision The document does not in any way claim property of the following content but serves as a material for only education purposes. Follow @AkhikoM on IG.  GenPhysics Notes 1st Quarter - Any interaction between particles, kinetic before and after the collision is aggregates of particles, or rigid bodies not equal in which they come near enough to exert a mutual influence, generally with exchange of energy. 3. Backward inelastic collision - Reverse inelastic collision where momentum is conserved and kinetic energy increases. - By Newton’s third law, the forces exerted onto each other are equal and opposite where the time in which they impact are the same. - Hence, Impulse of the 2 bodies are equal. Coefficient of restitution: - The ratio of final velocity to the initial 𝐹1 =− 𝐹2 𝑡1 = 𝑡2 (𝐹𝑡)1 =− (𝐹𝑡)2 𝐽1 =− 𝐽2 velocity between two objects after their Since the impulses are equal, the moments collision are also equal thus: - denoted as ‘e’ and is a unit less quantity, and its values range between 0 and 1. 𝑉𝑓𝑏−𝑉𝑓𝑎 - 𝑒= 𝑉𝑖𝑎−𝑉𝑖𝑏 Relationship between momentum and center of mass: For a collision to be linear, the force exerted upon an object must be within its center of mass else the object will rotate around its Types of Collision: center of mass. 1. Elastic Collision - collision that conserves kinetic energy. The kinetic energy before and after the collision is equal. Angular Momentum - rotational equivalent of linear momentum Hence the relationship of momentum and center of mass is that the momentum of a system must be in its center of mass for it to be linear else angular momentum will occur. 2. Inelastic Collision - collision that does not conserve kinetic energy. The The document does not in any way claim property of the following content but serves as a material for only education purposes. Follow @AkhikoM on IG.  GenPhysics Notes 1st Quarter Compilation of Formulas: Errors and Uncertainty Relative Motion Absolute 𝑉𝐴 − 𝑉 𝐸 𝑋𝑃𝐴 = 𝑋𝑃𝐵 + 𝑋𝐵𝐴 Error 𝑋𝐵𝐴 =− 𝑋𝐴𝐵 Relative 𝑉𝐴− 𝑉𝐸 Error 𝑉𝐸 Scalar and Vector Multiplication Variance 2 Σ(𝑥𝑖−µ) Σ(𝑥𝑖−𝑥) Scalar σ = 𝑛 𝑜𝑟 𝑛−1 ab = a * |b| Dot Product Scalar and Vector Quantities ab = |a| * |b| cosΘ Magnitude / 𝑉 = 2 𝑉𝑥 + 𝑉𝑦 2 Cross Product ab = |a| * |b| sinΘ Resultant Components Universal law of Gravitation x = R cos θ 𝐺 * 𝑚1*𝑚2 y = R sin θ 𝐹𝑔𝑎𝑣 = 2 −1 𝑦 𝑑 θ = 𝑡𝑎𝑛 | 𝑥 | Center of Mass Motion and Kinematics Ξ𝑚𝑥 𝐶𝑀 = Ξ𝑚 Displaceme ∆𝑥 = 𝑥𝑓 − 𝑥𝑖 nt Potential Energy Average ∆𝑥 𝑥𝑓−𝑥𝑖 Properties and 𝑊 = 𝑊 = 𝑊 𝑉= = 1 2 3 Velocity ∆𝑡 𝑡𝑓−𝑡𝑖 Theorems 𝑊𝐴𝐵 + 𝑊𝐵𝐴 = 0 Average ∆𝑣 𝑣𝑓−𝑣𝑖 𝑎= = 𝑊 = ∆𝐾𝐸 Acceleration ∆𝑡 𝑡𝑓−𝑡𝑖 𝐸 = 𝐾𝐸 + 𝑈 Uniformly Accelerated Motion / Projectile Energy Formulas 2 Horizontal 𝑉𝑓 = 𝑉0 + 𝑎𝑡 KE = ½ m𝑣 UAM U = mgh 1 2 ∆𝑥 = 𝑉0𝑡 + 𝑎𝑡 2 2 2 2 U = ½k ∆𝑥 𝑉𝑓−𝑉0 ∆𝑥 = 2𝑎 Impulse and Momentum 𝑉𝑓 +𝑉0 Impulse and Ξ𝐹 = ∆𝑃 ∆𝑥 = ( 2 )t Momentum ∆𝑡 P = mv Vertical Same as above but a = g; g = 9.8 Ft = mΔv UAM and m/s2 J = ∆𝑃 Y-Projectile X- Projectile ∆𝑥 = 𝑉 𝑡 𝐹𝑛𝑒𝑡 0 2 𝑉0 𝑠𝑖𝑛 2θ 𝑚 = 𝑎 ∆𝑥 = 𝑔 Δ𝑣 𝑎= 𝑡 Circular Motion 𝐹𝑛𝑒𝑡 Δ𝑣 𝑚 = 𝑡 Uniform 2 2 𝑉 4π 𝑟 𝑎𝑟𝑎𝑑 = = Ft = Δmv 𝑅 2 𝑡 2π𝑟 𝑉= 𝑡 Collision 𝐹1 =− 𝐹2 2 𝐹𝑐 = 𝑚𝑉 𝑡1 = 𝑡2 𝑟 (𝐹𝑡)1 =− (𝐹𝑡)2 Non-Uniform 2 2 𝑎= 𝑑|𝑣| = 𝑎𝑡𝑎𝑛 + 𝑎𝑅 = 𝑎𝑡𝑎𝑛 + 𝑎𝑅 𝐽1 =− 𝐽2 𝑑𝑡 Coefficient of 𝑉𝑓𝑏−𝑉𝑓𝑎 𝑒= Restitution 𝑉𝑖𝑎−𝑉𝑖𝑏 The document does not in any way claim property of the following content but serves as a material for only education purposes. 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