StudySmarter - The all-in-one study app.

4.8 • +11k Ratings

More than 3 Million Downloads

Free

Suggested languages for you:

Americas

Europe

Integrals of Motion

- 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
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits at Infinity and Asymptotes
- 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
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- 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
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- 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
- Angles in Polygons
- 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
- Binomial Theorem
- 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
- Deductive Reasoning
- 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
- Parallel Lines
- 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
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- 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
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- 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
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- 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
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Distribution Function
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Degrees of Freedom
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test for Regression Slope
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Paired T-Test
- Point Estimation
- Probability
- Probability Calculations
- Probability Density Function
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residual Sum of Squares
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Skewness
- Spearman's Rank Correlation Coefficient
- Standard Deviation
- Standard Error
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- T-distribution
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Variance for Binomial Distribution
- Venn Diagrams

When a car drives along a road, its position, velocity and acceleration will all change at various points in the trip. These three properties are all inherently linked: if the speed of the car changes, it must have accelerated or decelerated, and this will then cause the position of the car to change too.

Calculus is the study of change. For this reason, the use of calculus is an **integral **part of the study of motion, pun intended.

**Integration of motion** is a method of studying how objects move in space, through the use of integration. Before you cover this, it is important to recap some ideas to do with integration first.

Integration is a method that can be used to find the area under a graph.

The definite integral between \(a\) and \(b\) of the function above gives the area under the graph. If the function is \(f(t),\) this is written

\[ \int_a^b f(t) \, \textrm{d}t. \]

For more information on integrals, as well as how to find the integral of a certain function, see Integrals.

Displacement, velocity and acceleration are the most important factors when working with any sort of moving object. All of these definitions are assuming *an object is moving in one direction, along the real line*.

**Displacement** \(s(t)\) is how far away an object is from a certain position, taken as the origin.

Displacement is *directional*, meaning it can be *positive* or *negative* depending on which direction the object is from the origin.

**Velocity** \(v(t)\) is the rate of change of displacement.

- If the velocity is \(0,\) the object is
**stationary**. - If the velocity is
**positive**, the object is traveling*forward*(towards infinity on the real axis). - If the velocity is
**negative**, the object is traveling*backward*(towards negative infinity on the real axis).

**Acceleration** \(a(t)\) is the rate of change in velocity.

If the acceleration is 0, the object will travel at the

*same speed*.If the acceleration is

**positive**, the object will be*speeding up*if it is traveling forwards,or

*slowing down*if it is traveling backward.

If the acceleration is

**negative**, the object will be*slowing down*if it is traveling forwards,or

*speeding up*if it is traveling backward.

In the above definitions, \(t\) represents time. Since these definitions use rate of change, you can be sure that differentiation will be used when working with them. The following formulas hold:

\[ \begin{align} v(t) & = \frac{\mathrm{d}s}{\mathrm{d}t} \\ a(t) & = \frac{\mathrm{d}v}{\mathrm{d}t}. \end{align} \]

Given the functions \(s(t), v(t) \) and \( a(t),\) denoting displacement, velocity and acceleration, the following indefinite integral formulas exist:

\[ \begin{align} \int v(t) \, \mathrm{d}t & = s(t) + c \\ \int a(t) \, \mathrm{d}t & = v(t) + c. \end{align} \]

Don't forget to include your constant of integration when taking indefinite integrals.

This makes sense given the derivative relations in the previous section, as these are simply the inverse formulas. Let's look at some examples of using these formulas.

A particle is traveling with velocity \( v(t) = 6t^2, \) and has displacement \(100\) after 5 seconds. Find the velocity function of this particle.

**Solution:**

The first step is to take the indefinite integral of the particle.

\[ \begin{align} s(t) & = \int v(t) \, \mathrm{d}t \\ & = \int 6 t^2 \, \mathrm{d}t \\ & = 2 t^3 + c. \end{align} \]

Now, you can substitute \(t=5\) into the displacement vector to find the value of the constant of integration.

\[ \begin{align} v(5) = 100 & = 2 \cdot 5^3 + c \\ & = 2 \cdot 125 + c \\ & = 250 + c. \end{align} \]

Hence, you have that

\[ \begin{align} 250 + c & = 100 \\ \implies c & = -150. \end{align} \]

Hence, the displacement function must be

\[ s(t) = 2 t^3 - 150. \]

Now let's look at one where you must find the velocity function from the acceleration vector.

A particle is traveling with acceleration \( a(t) = \cos{t} \) and is stationary at \(t=0.\) Find the velocity function for this particle.

**Solution:**

First, integrate the acceleration function.

\[ \begin{align} v(t) & = \int a(t) \, \mathrm{d}t \\ & = \int \cos{t} \, \mathrm{d}t \\ & = \sin{t} + c. \end{align} \]

Now, since the particle is stationary at \(t=0,\) the velocity must be \(0.\) Substitute this in to find the value of \(c.\)

\[ v(0) = 0 = \sin(0) + c = c \]

Hence, \(c\) must be \(0.\) This means the final velocity function must be

\[ v(t) = \sin{t}. \]

Definite integrals can be used to calculate the total displacement between two times \(t=a\) and \(t=b\). The **total displacement** is the distance between the position of the particle at \(t=a\) and \(t=b\). This can be found by taking the definite integral of the particle with bounds \(a\) and \(b\):

\[ \text{total displacement} = \int_a^b v(t) \, \mathrm{d}t. \]

You can also find the **total distance** that the particle has traveled during this time, by taking the absolute value of the velocity before integrating it.

\[ \text{total distance} = \int_a^b |v(t)| \, \mathrm{d}t. \]

The easiest way to see the difference between total displacement and total distance is an example.

If you throw a ball \(2\) meters into the air and catch it, the total displacement is \(0\) because it has returned to its original spot. The total distance, however, is \(4\) meters, since it traveled \(2\) meters into the air and \(2\) meters back down. If the object is always traveling with positive velocity, the total displacement and total distance traveled will always be the same.

Let's look at an example of finding the displacement and distance of a particle.

A particle travels with velocity \(v(t) = 8 -4t.\) Find the total displacement and the total distance of the particle between times \(t=1\) and \(t=3.\)

**Solution:**

First, find the total displacement. Take the definite integral of the velocity function between \(t=1\) and \(t=3.\)

\[ \begin{align} \int_1^3 8 - 4t \, \mathrm{d}t & = [8t - 2t^2]_1^3 \\ & = (8 \cdot 3 - 2 \cdot 3^2) - (8 \cdot 1 - 2 \cdot 1^2) \\ & = (24 - 18) - (8 - 2) \\ & = 6 - 6 \\ & = 0. \end{align} \]

So the total displacement is \(0.\)

To find the total distance travelled, you must find the integral of the absolute value of the function.

\[ \int_1^3 | 8 - 4t | \, \mathrm{d}t. \]

The function will go negative at \(t=2,\) hence the integral can be split into two at this point.

\[ \begin{align} \int_1^3 | 8 - 4t | \, \mathrm{d}t & = \int_1^2 8 - 4t \, \mathrm{d}t + \int_1^2 -(8-4t) \, \mathrm{d}t \\ & = \int_2^3 8 - 4t \, \mathrm{d}t + \int_1^2 -8+4t \, \mathrm{d}t. \end{align} \]

Now you can evaluate the integrals on the left to deduce the total distance travelled.

\[ \begin{align} \int_1^3 | 8 - 4t | \, \mathrm{d}t & = [8t - 2t^2]_1^2 + [-8t + 2t^2]_2^3 \\ & = [(8\cdot 2 - 2\cdot 2^2) - (8 \cdot 1 - 2 \cdot 1^2)] + [(-8\cdot 3 + 2 \cdot 3^2) - (-8 \cdot 2 - 2 \cdot 2^2)] \\ & = [(16 - 8) + (8 - 2)] + [(-24 + 18) - (-16 + 8)] \\ & = [8 - 6] + [-6 + 8] \\ & = 4. \end{align} \]

So the total distance travelled is \(4.\)

If a particle is travelling with velocity \(v_1(t)\) for the first \(a\) seconds and then travels at velocity \(v_2(t)\) for the following \(b\) seconds, the displacement can be found by adding these two integrals together.

\[ \text{total displacement} \int_0^a v_1(t) \, \mathrm{d}t + \int_a^b v_2(t) \, \mathrm{d}t. \]

This same technique can be used for any finite number of separate velocity functions too, and can be used to find the total distance travelled by integrating the absolute values of the velocities instead.

A particle travels with velocity \(v_1(t) = t^2 \) for 3 seconds, and then has velocity \(v_2(t) = 3t\) for a further 6 seconds. Calculate the total distance travelled by the particle in this time.

**Solution:**

First, since the question asks for the total distance travelled, you must take the absolute values of the velocities. But since both of the velocities are always positive, the total distance travelled will be equal to the displacement, and the absolute values are irrelevant. Hence, the total distance travelled is

\[ \begin{align} \int_0^3 3t^2 \, \textrm{d}t + \int_3^9 3t \, \textrm{d}t & = [t^3]_0^3 + \left[\frac{3}{2} t^2 \right]_3^9 \\ & = (3^3 - 0^3) + \left(\frac{3}{2} \cdot 9^2 - \frac{3}{2} \cdot 3^2 \right) \\ & = 27 + (\frac{243}{2} - \frac{27}{2}) \\ & = 27 + 116 \\ & = 143 \end{align} \]

So the total displacement and total distance travelled are 143 units.

Let's look at some more complicated examples of questions using integration and motion.

The acceleration of a particle is given by \[a(t) = 6t\] and the position of the particle is \(10\) at \(t=0\) and \(14\) at \(t=2.\) calculate the displacement function of the particle.

**Solution:**

This time, you will have to integrate twice, since the function given is acceleration and not velocity. Integrating the function once will give you:

\[ \begin{align} v(t) & = \int a(t) \, \textrm{d}t \\ & = \int 6t \, \textrm{d}t \\ & = 3t^2 + c. \end{align}\]

Now, you can integrate the velocity function to find the displacement function.

\[ \begin{align} s(t) & = \int v(t) \, \textrm{d}t \\ & = \int 3 t^2 + c \, \textrm{d}t \\ & = t^2 + ct + d. \end{align} \]

\(d\) is another constant of integration. Now, you can substitute in the initial conditions to find the exact displacement function. If \(t=0:\)

\[ \begin{align} s(0) = 10 & = 0^2 + c \cdot 0 + d \\ \implies d & = 10. \end{align} \]

Hence, \(d\) must be \(0.\) If \(t = 2:\)

\[ \begin{align} s(2) = 14 & = 2^2 + c \cdot 10 + 10 \\ & = 14 + 10c \\ \implies 0 & = 10c \end{align} \]

so \(c = 0.\) Hence, the final formula for displacement is

\[ s(t) = t^2 + 10. \]

Now let's look at another example of a question that may be a little more difficult.

Particle \(A\) is travelling at velocity \(v_A(t) = 2t^2 \) and particle \(B\) is travelling at velocity \(v_B(t) = 4t.\) Both particles are in the same position at time \(t=0.\)

- Find the distance between the particles at time \(t=5.\)
- At what time are the particles in the same position again?

**Solution:**

1) Since the particles are only moving in one dimension, the distance between them will simply be the absolute value of the difference between their distances from the starting points.

\[ \begin{align} \text{Distance between A and B at time 5} & = \int_0^5 v_A(t) \, \textrm{d}t - \int v_B(t) \, \textrm{d}t \\ & = \int v_A(t) - v_B(t) \, \textrm{d}t. \end{align} \]

Now, you can plug the formulas for \(v_A(t)\) and \(v_B(t)\) into the formula above to get the distance between them.

\[ \begin{align} \int_0^5 v_A(t) - v_B(t) \, \textrm{d}t & = \int_0^5 2t^2 - 4t \, \textrm{d}t \\ & = \left[ \frac{2}{3} t^3 - 2 t^2 \right]_0^5 \\ & = \frac{2}{3} \cdot 5^3 - 2 \cdot 5^2 \\ & = \frac{250}{3} - 50 \\ & = \frac{100}{3}. \end{align} \]

So the distance between the two particles at time \(t=0\) is \(\frac{100}{3}.\)

2) This problem can be solved in the same way as question one, but you must use a variable, call it \(t',\) as the upper bound of integration, then set the result as equal to \(0\) and solve the equation for \(t'.\) Using the same steps as before, you will get that:

\[ \begin{align} \int_0^5 v_A(t) - v_B(t) \, \textrm{d}t & = \int_0^5 2t^2 - 4t \, \textrm{d}t \\ & = \left[ \frac{2}{3} t^3 - 2 t^2 \right]_0^{t'} \\ & = \frac{2}{3} t'^3 - 2 t'^2. \end{align} \]

Now you can set this equal to 0, and solve for \(t'.\)

\[ \begin{align} \frac{2}{3} t'^3 - 2 t'^2 & = 0 \\ & \implies t' (\frac{2}{3} t' - 2 ) = 0. \end{align} \]

For this equation to be true, either \(t'=0\) or \(\frac{2}{3} t' - 2 = 0.\) \(t'=0\) is simply the starting point, and this is not the answer that the question is looking for. Hence, it must be that

\[ \begin{align} \frac{2}{3} t' - 2 & = 0 \\ \implies t' & = 2 \cdot \frac{3}{2} \\ \implies t' & = 3\ \end{align} \]

Hence, the particles will be in the same position again at \(t=3.\)

Integrals of motion are essential in areas of physics such as classical mechanics and general relativity. These relations between position, velocity and acceleration allow scientists and engineers to model anything that moves such as cars, rockets and objects in space. Of course, the equations used in these models tend to be much more complicated than just polynomials or trigonometric functions.

- The following formulas exist, relating displacement \(s(t),\) velocity \(v(t),\) and acceleration \(a(t)\) together \[ \begin{align} \int v(t) \, \mathrm{d}t & = s(t) + c \\ \int a(t) \, \mathrm{d}t & = v(t) + c. \end{align} \]
- The total displacement between times \(a\) and \(b\) of an object is the difference between it's start and end points. The formula for finding total displacement from velocity \(v(t)\) is \[ \text{total displacement} = \int_a^b v(t) \, \mathrm{d}t. \]
- The total distance travelled between times \(a\) and \(b\) of an object is how far the object travelled during this time, irrespective of direction. \[ \text{total distance} = \int_a^b |v(t)| \, \mathrm{d}t. \]
- If the velocity is always constant, total displacement and total distance travelled will always be equal.

The integral of acceleration is velocity, and the integral of velocity is displacement.

The integral of acceleration is velocity, and the integral of velocity is displacement.

The integration of velocity is displacement.

More about Integrals of Motion

Be perfectly prepared on time with an individual plan.

Test your knowledge with gamified quizzes.

Create and find flashcards in record time.

Create beautiful notes faster than ever before.

Have all your study materials in one place.

Upload unlimited documents and save them online.

Identify your study strength and weaknesses.

Set individual study goals and earn points reaching them.

Stop procrastinating with our study reminders.

Earn points, unlock badges and level up while studying.

Create flashcards in notes completely automatically.

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.

Over 10 million students from across the world are already learning smarter.

Get Started for Free