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Kinematics

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Area Between Two Curves
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
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- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
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- Jump Discontinuity
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- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Radius of Convergence
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Separation of Variables
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
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- Area and Volume
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- Area of Trapezoid
- Area of a Kite
- Composition
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- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
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- Glide Reflections
- HL ASA and AAS
- Identity Map
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- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
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- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
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- Volume of Cone
- Volume of Cylinder
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- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Conservation of Mechanical Energy
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Elastic Strings and Springs
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Power
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Work Done by a Constant Force
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
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- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Argand Diagram
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
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- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- De Moivre's Theorem
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- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
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- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
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- Proof by Exhaustion
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- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
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- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
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- Representation of Complex Numbers
- Rewriting Formulas and Equations
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- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
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- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
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- Simplifying Fractions
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- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
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- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
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- Special Products
- Standard Form
- Standard Integrals
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- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
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- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
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- Unit Circle
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- Variables in Algebra
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- Verifying Trigonometric Identities
- Writing Equations
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- Statistics
- Bias in Experiments
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- Bivariate Data
- Box Plots
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- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
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- Comparing Two Means Hypothesis Testing
- Conditional Probability
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- Confidence Intervals
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- Data Analysis
- Data Interpretation
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- Frequency Polygons
- Generalization and Conclusions
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- Hypothesis Test for Correlation
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- Inference for Distributions of Categorical Data
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Kinematics focuses on the movement of objects. It deals with forces and the geometric aspects of motion, which is related to velocity and acceleration.

Let's define some specific terms to understand the topic of kinematics:

Distance is the total length covered irrespective of the direction of a motion.

Displacement is the distance moved in a certain direction.

Speed is the distance covered per unit of time.

Velocity is the rate of change of displacement.

Acceleration is the rate of change of velocity.

We can now use these to describe the motion of an object and the geometry of the curve it creates as it moves.

In variable acceleration, acceleration changes over time. We need to use differentiation and integration to convert between displacement, velocity, and acceleration. We use differentiation to convert from displacement to velocity and from velocity to acceleration. Then we use integration to go back from acceleration to velocity and from velocity to displacement.

Fig: Variable acceleration

Find the times at which a particle is at instantaneous rest if its displacement is given as

Answer:

Use differentiation to convert from displacement into velocity.

This is the velocity function. We want to know when v = 0

(3t -2) (t -4) = 0

The particle is at instantaneous at t = 4s or t = 2/3 s.

Calculus makes deriving two of the three equations of motion easy. Since we know acceleration is the first derivative of velocity with respect to time, use the definition to reverse it. Instead of finding velocity, integrate it to find acceleration.

Various functions describe the quantities of acceleration, displacement, and velocity. They can also be transformed into functions that describe the other two quantities. There are two ways of doing this: differentiation (finding the derivative) or integration (finding the integral).

Displacement s = f (t) [location in reference to an origin]

Velocity [derivative of displacement]

Acceleration [derivative of velocity]

Let's analyze these concepts together. You can understand the characteristics of kinematics based on the output of our function. For displacement, a value of 0 means our position is at the origin. If displacement is positive, then it's to the right of the origin, and if it is negative, then it's to the left of the origin.

Velocity has quite a similar interpretation. If a value is 0, it means we are at rest. Positive velocity means movement to the right, and negative velocity means movement to the left.

A 0 output for the acceleration function means velocity is either at a maximum, a minimum, or constant. Positive output means velocity is increasing, and negative output means velocity is decreasing.

Find the velocity if the displacement is given as a function of time .

Answer:

We will simply differentiate this:

This is the same as [Look at differentiation if struggling]

v = 6t

If you wanted to find the acceleration, this could be written as the second derivative of displacement with respect to time or the velocity differentiated with respect to time.

a = 6

So this particle has a constant acceleration of 6.

Formally, this is known as 'one-dimensional equations of motion for constant acceleration'. Kinematic equations in this section are only valid when acceleration is constant, and motion is constrained to a straight line.

As mentioned earlier, acceleration is constant, meaning it is the same regardless of time. We use standard notation here, with a representing acceleration, u representing initial velocity, and t representing time.

Let's draw graphs for all three properties of motion: displacement, velocity, and acceleration. We will identify the motion objects from the graphs.

**Acceleration time graph:**

Here's the scenario: a bullet is fired vertically. The only force acting on it is gravity. Gravity doesn't change; therefore, the acceleration doesn't change either. Below is a demonstration of the acceleration time graph.

Acceleration

**Velocity time graph:**

Since we know that the acceleration is constant and negative because it is acting in the opposite direction to the velocity, that will mean the gradient of the velocity-time graph should be constant and negative too, which will give us something like this:

Velocity

**Displacement time graph:**

The graph below shows the bullet from its start point until it reaches maximum height, where the velocity is 0 before it starts coming down again. You have got a gradient that is constant and negative. So a graph for displacement will look like this:

Displacement

Note that velocity is always positive but decreases to 0. Therefore the gradient of the displacement vs time graph should always be positive and should decrease to 0.

The SUVAT equations are a set of equations that describe motion in one dimension, each with one variable omitted.

variable | Description | SI unit |

s | Displacement | meters (m) |

u | Initial velocity | m / s (meters per second) |

v | Final velocity | m / s (meters per second) |

a | Acceleration | m / s / s (meters per second per second) |

t | Time | s (seconds) |

First equation: v = u + at

Second equation:

Third equation:

Forth equation:

A ferry carries passengers between the banks of a river which are 20m apart. After setting off, the ferry accelerates at 0.2 for 12 seconds before turning off the engine and decelerating at a constant rate and coming to a stop at the opposite bank.

Calculate the speed of the ferry after 12 seconds.

Calculate the distance the ferry travels during these 12 seconds.

Answer:

Write your SUVAT down, clarify the variables you have and those you don't, and work out what equation to use.

s = x m

u = 0 m / s

v =?

a = 0.2

t = 12 s

Now we find the SUVAT equation which does not include s.

v = u + at

v = 0 + 0.2 12

v = 2.4 m / s

Using

s = 14.4 m

Projectiles deal with objects that are projected through the air either by being thrown, hit, fired, etc. Examples are:

Someone throws a ball.

A bullet at the instant it is fired.

A car driving off a cliff.

Projectiles

A trajectory is the path of a projectile. A projectile always has some initial speed, usually angled to the horizontal. It moves freely under gravity because the only force acting on it is its weight. To solve a projectile problem, you need to use a SUVAT equation in two dimensions to split the horizontal and vertical components of each value.

Since trigonometric functions are used to find unknown angles and distances, we will use that to resolve the components here.

A particle P is projected from a point O on a horizontal plane with a speed of 28m / s at 30 degrees to the horizontal. After projection, the particle moves freely under gravity until it strikes the plane at point A. Find the greatest height of the particle, the time of flight, and the distance OA.

Answer:

We want to find a vertical height so we will use SUVAT in the vertical direction.

s =?

u = 28sin30 = 14 m / s

v = 0

a = -g

t = x

With a situation like this, we use an equation that doesn't have t.

If we plug in the values, we will get 0 = 196 - 2gs

And we take g to be equal to 9.8, s = 10m

The second thing we have to find is the time of flight (the time it takes P to get from O to A).

s = 0 m

u = 14 m / s

v = x

a = -g

t =?

So, t = 0 is when particle P is at the origin.

Whereas the solution where is when the particle is at A.

And if we take g = 9.8, t = 2.9s.

For the final part of the problem, we need to find the distance between O and A.

We are going to find a horizontal distance, so we will use SUVAT in the horizontal direction.

s =?

u = 28cos30 m / s

v = x

a = 0

The distance between O and A is s = 69 m

- Displacement is the change in the position of an object relative to its frame of reference.
- Acceleration due to gravity is either 9.8 on the surface of the earth. It can be considered as 10 for the ease of calculation.
- You can use calculus techniques to analyze functions too.
- To solve kinematic problems in two dimensions, you need to use geometry to determine unknown magnitudes and directions and use the directions to determine kinematic quantities.

Kinematics is an area of study that focuses on the movement of objects.

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