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average and instantaneous acceleration of an object

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Interface

#include <codecogs/physics/kinematics/acceleration.h>

using namespace Physics::Kinematics;

inline double acceleration_avg(double vf, double tf, double v0 = 0, double t0 = 0)
inline double acceleration_ins(double (*v)(double), double t, double eps = 1E-6)
inline std::vector<double> acceleration_ins(double (*v)(double), std::vector<double> &t, double eps = 1E-6)
inline double acceleration_ins_space(double (*x)(double), double t, double eps = 1E-5)
inline std::vector<double> acceleration_ins_space(double (*x)(double), std::vector<double> &t, double eps = 1E-5)

Overview

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

FUNCTION

acceleration_avg

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

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

(1)

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

final velocity (meters per second)

final time (seconds) [needs to be different from t0]

Default value = 0

Returns

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

Interactive Calculator

Result –

FUNCTION

acceleration_ins

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 $a(t)$ is given by the derivative of the velocity function $v(t)$ with respect to time, i.e.

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

(2)

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 1

#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

function defining the velocity of the object at any moment of time (meters per second)

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)

FUNCTION

acceleration_ins

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 1

#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

function defining the velocity at any moment of time (meters per second)

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)

FUNCTION

acceleration_ins_space

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 $a(t)$ is given by the second order derivative of the position function $x(t)$ with respect to time, i.e.

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

(3)

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 1

#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

function defining the position of the object at any moment of time (meters)

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)

FUNCTION

acceleration_ins_space

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

Example 1

#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

function defining the position at any moment of time (meters)

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)