Physics › Kinematics ›

# acceleration

Average and instantaneous acceleration of an object

Controller: **CodeCogs**

**Contents**

## Dependents

## Interface

C++

Excel

## Overview

This module computes the average and instantaneous acceleration of a moving object at given moments of time.### Authors

*Lucian Bentea (July 2007)*

## Acceleration Avg

doubleacceleration_avg( | double | vf | |

double | tf | ||

double | v0` = 0` | ||

double | t0` = 0` | )[inline] |

### Example 1

#include <codecogs/physics/kinematics/acceleration.h> #include <iostream> int main() { // final velocity and time double v = 200, t = 15.7; std::cout << std::endl; std::cout << " Final velocity = " << v << " m/s" << std::endl; std::cout << " Time spent = " << t << " s" << std::endl; std::cout << std::endl; // assuming initial velocity and initial time are null, // display the average acceleration of the object std::cout << "Average acceleration = " << Physics::Kinematics::acceleration_avg(v, t); std::cout << " m/s^2" << std::endl; return 0; }

**Output**Final velocity = 200 m/s Time spent = 15.7 s Average acceleration = 12.7389 m/s^2

### Parameters

vf final velocity (meters per second) tf final time (seconds) [needs to be different from t0] v0 Default value = 0 t0 Default value = 0

### Returns

- the average acceleration of the moving object (meters per sq. second)

##### Source Code

This module is private, for owner's use only.

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## Acceleration Ins

doubleacceleration_ins( | double | (*v)(double)[function pointer] | |

double | t | ||

double | eps` = 1E-6` | )[inline] |

*t*, given a function

*v*which describes the velocity of the object at any moment of time. It is based on the fact that the instantaneous acceleration function is given by the derivative of the velocity function with respect to time, i.e. Since this function uses numerical differentiation to compute the above derivative, an optional parameter

*eps*is available to specify the precision of numerical computations.

### Example 2

#include <codecogs/physics/kinematics/acceleration.h> #include <iostream> // function defining the velocity at any moment of time t; // in this case velocity(t) = t^2/2 double velocity(double t) { return t*t/2; } int main() { // time at which to calculate instantaneous acceleration double t = 11.43; std::cout << std::endl; std::cout << "Velocity = " << velocity(t); std::cout << " m/s" << std::endl; std::cout << " Time = " << t; std::cout << " s" << std::endl; std::cout << std::endl; // display instantaneous acceleration at time t std::cout << "Instantaneous acceleration = " << Physics::Kinematics::acceleration_ins(velocity, t); std::cout << " m/s^2" << std::endl; return 0; }

**Output**Velocity = 65.3225 m/s Time = 11.43 s Instantaneous acceleration = 11.43 m/s^2

### Parameters

v function defining the velocity of the object at any moment of time (meters per second) t the moment of time at which the instantaneous acceleration is to be evaluated (seconds) eps Default value = 1E-6

### Returns

- the instantaneous acceleration of the object at time
*t*(meters per sq. second)

##### Source Code

This module is private, for owner's use only.

Not a member, then Register with CodeCogs. Already a Member, then Login.

## Acceleration Ins

std::vector<double>acceleration_ins( | double | (*v)(double)[function pointer] | |

std::vector<double>& | t | ||

double | eps` = 1E-6` | )[inline] |

### Example 3

#include <codecogs/physics/kinematics/acceleration.h> #include <iostream> // function defining the velocity at any moment of time t; // in this case velocity(t) = t^2/2 double speed(double t) { return t*t/2; } int main() { // moments of time at which to evaluate // the instantaneous acceleration of the object double t[10] = { 11.40, 11.41, 11.42, 11.43, 11.44, 11.45, 11.46, 11.47, 11.48, 11.49 }; // compute the instantaneous acceleration values std::vector<double> time(t, t+10), acceleration = Physics::Kinematics::acceleration_ins(speed, time); // display the time, the velocity // and the instantaneous acceleration std::cout << std::endl; for (int i = 0; i < 10; i++) { std::cout << "Time = " << time[i] << " s"; std::cout << "\tVelocity = " << speed(time[i]) << " m/s"; std::cout << "\tAcceleration = " << acceleration[i] << " m/s^2"; std::cout << std::endl; } return 0; }

**Output**Time = 11.4 s Velocity = 64.98 m/s Acceleration = 11.4 m/s^2 Time = 11.41 s Velocity = 65.094 m/s Acceleration = 11.41 m/s^2 Time = 11.42 s Velocity = 65.2082 m/s Acceleration = 11.42 m/s^2 Time = 11.43 s Velocity = 65.3225 m/s Acceleration = 11.43 m/s^2 Time = 11.44 s Velocity = 65.4368 m/s Acceleration = 11.44 m/s^2 Time = 11.45 s Velocity = 65.5512 m/s Acceleration = 11.45 m/s^2 Time = 11.46 s Velocity = 65.6658 m/s Acceleration = 11.46 m/s^2 Time = 11.47 s Velocity = 65.7805 m/s Acceleration = 11.47 m/s^2 Time = 11.48 s Velocity = 65.8952 m/s Acceleration = 11.48 m/s^2 Time = 11.49 s Velocity = 66.0101 m/s Acceleration = 11.49 m/s^2

### Parameters

v function defining the velocity at any moment of time (meters per second) t array containing the moments of time at which the instantaneous acceleration should be evaluated (seconds) eps Default value = 1E-6

### Returns

- array containing the instantaneous acceleration values at the moments of time given by
*t*(meters per sq. second)

##### Source Code

This module is private, for owner's use only.

Not a member, then Register with CodeCogs. Already a Member, then Login.

## Acceleration Ins Space

doubleacceleration_ins_space( | double | (*x)(double)[function pointer] | |

double | t | ||

double | eps` = 1E-5` | )[inline] |

*t*, given a function

*x*which determines the position of the object at any moment of time on a fixed axis. It is based on the fact that the instantaneous acceleration function is given by the second order derivative of the position function with respect to time, i.e. Since this function uses numerical differentiation to compute the above second order derivative, an optional parameter

*eps*is available to specify the precision of numerical computations.

### Example 4

#include <codecogs/physics/kinematics/acceleration.h> #include <iostream> // function defining the position at any moment of time t; // in this case space(t) = t^3 double pos(double t) { return t*t*t; } int main() { // time at which to calculate instantaneous acceleration double t = 11.43; std::cout << std::endl; std::cout << "Position = " << space(t); std::cout << " m" << std::endl; std::cout << " Time = " << t; std::cout << " s" << std::endl; std::cout << std::endl; // display instantaneous acceleration at time t std::cout << "Instantaneous acceleration = " << Physics::Kinematics::acceleration_ins_space(space, t); std::cout << " m/s^2" << std::endl; return 0; }

**Output**Position = 1493.27 m Time = 11.43 s Instantaneous acceleration = 68.5782 m/s^2

### Parameters

x function defining the position of the object at any moment of time (meters) t the moment of time at which the instantaneous acceleration is to be evaluated (seconds) eps Default value = 1E-5

### Returns

- the instantaneous acceleration of the object at time
*t*(meters per sq. second)

##### Source Code

This module is private, for owner's use only.

Not a member, then Register with CodeCogs. Already a Member, then Login.

## Acceleration Ins Space

std::vector<double>acceleration_ins_space( | double | (*x)(double)[function pointer] | |

std::vector<double>& | t | ||

double | eps` = 1E-5` | )[inline] |

### Example 5

#include <codecogs/physics/kinematics/acceleration.h> #include <iostream> // function defining the position at any moment of time t; // in this case pos(t) = t^3 double pos(double t) { return t*t*t; } int main() { // moments of time at which to evaluate // the instantaneous acceleration of the object double t[10] = { 11.40, 11.41, 11.42, 11.43, 11.44, 11.45, 11.46, 11.47, 11.48, 11.49 }; // compute the instantaneous acceleration values std::vector<double> time(t, t+10), acceleration = Physics::Kinematics::acceleration_ins_space(distance, time); // display the time, the position // and the instantaneous acceleration values std::cout << std::endl; for (int i = 0; i < 10; i++) { std::cout << "Time = " << time[i] << " s"; std::cout << "\tPosition = " << distance(time[i]) << " m"; std::cout << "\tAcceleration = " << acceleration[i] << " m/s^2"; std::cout << std::endl; } return 0; }

**Output**Time = 11.4 s Position = 1481.54 m Acceleration = 68.3985 m/s^2 Time = 11.41 s Position = 1485.45 m Acceleration = 68.4599 m/s^2 Time = 11.42 s Position = 1489.36 m Acceleration = 68.5168 m/s^2 Time = 11.43 s Position = 1493.27 m Acceleration = 68.5782 m/s^2 Time = 11.44 s Position = 1497.19 m Acceleration = 68.6396 m/s^2 Time = 11.45 s Position = 1501.12 m Acceleration = 68.6987 m/s^2 Time = 11.46 s Position = 1505.06 m Acceleration = 68.7601 m/s^2 Time = 11.47 s Position = 1509 m Acceleration = 68.8237 m/s^2 Time = 11.48 s Position = 1512.95 m Acceleration = 68.8829 m/s^2 Time = 11.49 s Position = 1516.91 m Acceleration = 68.9397 m/s^2

### Parameters

x function defining the position at any moment of time (meters) t array containing the moments of time at which the instantaneous acceleration values should be evaluated (seconds) eps Default value = 1E-5

### Returns

- array containing the instantaneous acceleration values of the object at moments of time given by
*t*(meters per sq. second)

##### Source Code

This module is private, for owner's use only.

Not a member, then Register with CodeCogs. Already a Member, then Login.