CodeCogs

average and instantaneous velocity of an object

View versions (1)

Interface

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

using namespace Physics::Kinematics;

inline double velocity_avg(double xf, double tf, double x0 = 0, double t0 = 0)
inline double velocity_ins(double (*x)(double), double t, double eps = 1E-6)
inline std::vector<double> velocity_ins(double (*x)(double), std::vector<double> &t, double eps = 1E-6)

Overview

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

FUNCTION

velocity_avg

This function computes the average velocity of a moving object, given its initial and final position on an axis and also the total time spent to get to that final position. Considering $x_0$ is the initial position at time $t_0$ and $x_f$ is the final position at time $t_f$ , the average velocity $\overline{v}$ is given by the following simple formula:

$\overline{v} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_0}{t_f - t_0}.$

(1)

Example 1

#include <codecogs/physics/kinematics/velocity.h>
#include <iostream>

int main()
{
  // final position and time
  double x = 100, t = 15.7;

  std::cout << std::endl;
  std::cout << "  Final position = " << x << " m" << std::endl;
  std::cout << "      Time spent = " << t << " s" << std::endl;
  std::cout << std::endl;
 
  // assuming initial position and initial time are null,
  // display the average velocity of the object

  std::cout << "Average velocity = " << 
  Physics::Kinematics::velocity_avg(x, t);
  std::cout << " m/s" << std::endl;
 
  return 0;
}

Output

Final position = 100 m
      Time spent = 15.7 s

Average velocity = 6.36943 m/s

Parameters

final position on the axis (meters)

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

Default value = 0

Returns

the average velocity of the moving object (meters per second)

Interactive Calculator

Result –

FUNCTION

velocity_ins

This function returns the instantaneous velocity 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. It is based on the fact that the instantaneous velocity function $v(t)$ is given by the derivative of the position function $x(t)$ with respect to time, i.e.

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

(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/velocity.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()
{
  // time at which to calculate instantaneous velocity
  double t = 11.43;

  std::cout << std::endl;
  std::cout << "Position = " << pos(t);
  std::cout << " m" << std::endl;
  std::cout << "    Time = " << t;
  std::cout << " s" << std::endl;
  std::cout << std::endl;

  // display instantaneous velocity at time t

  std::cout << "Instantaneous velocity = " <<
  Physics::Kinematics::velocity_ins(space, t);
  std::cout << " m/s" << std::endl;
  
  return 0;
}

Output

Position = 1493.27 m
    Time = 11.43 s

Instantaneous velocity = 391.935 m/s

Parameters

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

the moment of time at which the instantaneous velocity is to be evaluated (seconds)

eps

Default value = 1E-6

Returns

the instantaneous velocity of the object at time t (meters per second)

FUNCTION

velocity_ins

This function is based on the same equation as the previous one, only that it is able to compute the instantaneous velocities 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/velocity.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 velocity 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 velocities

  std::vector<double> time(t, t+10),
  velocities = Physics::Kinematics::velocity_ins(distance, time);

  // display the time, the position
  // and the instantaneous velocities

  std::cout << std::endl;
  for (int i = 0; i < 10; i++)
  {
    std::cout << "Time = " << time[i] << " s";
    std::cout << "\tPosition = " << pos(time[i]) << " m";
    std::cout << "\tVelocity = " << velocities[i] << " m/s";
    std::cout << std::endl;
  }

  return 0;
}

Output

Time = 11.4 s   Position = 1481.54 m    Velocity = 389.88 m/s
Time = 11.41 s  Position = 1485.45 m    Velocity = 390.564 m/s
Time = 11.42 s  Position = 1489.36 m    Velocity = 391.249 m/s
Time = 11.43 s  Position = 1493.27 m    Velocity = 391.935 m/s
Time = 11.44 s  Position = 1497.19 m    Velocity = 392.621 m/s
Time = 11.45 s  Position = 1501.12 m    Velocity = 393.307 m/s
Time = 11.46 s  Position = 1505.06 m    Velocity = 393.995 m/s
Time = 11.47 s  Position = 1509 m       Velocity = 394.683 m/s
Time = 11.48 s  Position = 1512.95 m    Velocity = 395.371 m/s
Time = 11.49 s  Position = 1516.91 m    Velocity = 396.06 m/s

Parameters

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

array containing the moments of time at which the instantaneous velocities should be evaluated (seconds)

eps

Default value = 1E-6

Returns

array containing the instantaneous velocities of the object at moments of time given by t (meters per second)