Physics › Kinematics ›

acceleration

Average and instantaneous acceleration of an object

Controller: CodeCogs

Contents

Interface
Acceleration Avg
Acceleration Ins
Acceleration Ins
Acceleration Ins Space
Acceleration Ins Space
Page Comments

Private project under development, to help contact the author:

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]

This function computes the average acceleration of a moving object, given the initial and final velocities, and also the total time spent. Considering $\inline v_0$ is the initial velocity at time $\inline t_0$ and $\inline v_f$ is the final velocity at time $\inline t_f$ , the average acceleration $\inline \overline{a}$ is given by the following simple formula:

$\overline{a} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_0}{t_f - t_0}.$

(2)

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.

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

Acceleration Ins

doubleacceleration_ins(	double	(`v`)(double)[function pointer]*
	double	`t`
	double	`eps` `= 1E-6`	)[inline]

This function returns the instantaneous acceleration of a moving object at a certain moment of time 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 $\inline a(t)$ is given by the derivative of the velocity function $\inline v(t)$ with respect to time, i.e.

$a(t) = \frac{\mathrm{d}v}{\mathrm{d}t}(t).$

(3)

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]

This function is based on the same equation as the previous one, only that it is able to compute the instantaneous acceleration values at several moments of time and return the results in the form of an array.

Notice the example code below which shows exactly how this is a generalisation of the previous overloaded function.

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]

This function returns the instantaneous acceleration of a moving object at a certain moment of time 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 $\inline a(t)$ is given by the second order derivative of the position function $\inline x(t)$ with respect to time, i.e.

$a(t) = \frac{\mathrm{d}^2x}{\mathrm{d}t^2}(t).$

(4)

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]

Notice the example code below which shows how this is a generalisation of the previous overloaded function.

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.

double	`acceleration_avg` (double vf, double tf, double v0 = 0, double t0 = 0)[inline] Average acceleration of an object given the initial and final velocities, and also the time spent
double	`acceleration_ins` (double (v)(double), double t, double eps = 1E-6)[inline]* Instantaneous acceleration of an object at a certain moment of time, given the velocity function
std::vector<double>	`acceleration_ins` (double (v)(double), std::vector<double> &t, double eps = 1E-6)[inline]* Instantaneous acceleration values of an object at several moments of time, given the velocity function
double	`acceleration_ins_space` (double (x)(double), double t, double eps = 1E-5)[inline]* Instantaneous acceleration of an object at a certain moment of time, given the position function
std::vector<double>	`acceleration_ins_space` (double (x)(double), std::vector<double> &t, double eps = 1E-5)[inline]* Instantaneous acceleration values of an object at several moments of time, given the position function