If you throw something vertically upward and could somehow eliminate or ignore the effect of drag and air on the object , Then the object accelerates constantly when it goes up and falls down , This is called free fall.

If something falls freely under the effect of earth’s gravity without any effect of air then the phenomenon is called free fall.

While the free fall , no matter how big , small or weighty the object is ,  every object feel the same constant acceleration , the constant acceleration during free fall is called free fall acceleration.

The free fall acceleration of earth is denoted by “g” and it’s value at the surface of earth is approximately , But you should also note that the value of “g” varies slightly with change in latitude and elevation from surface of earth.

Free fall acceleration  equations:

Free fall acceleration is a constant acceleration so we can analyze the situations of free fall acceleration with a set of equations which are similar to constant acceleration equations.

We can use the constant acceleration equations to analyze the free fall with some changes in the equation.

We can suppose the motion in vertically upward direction as positive motion and the motion in downward direction as negative motion.

So we can replace the “acceleration = a”  in constant acceleration equations by “free fall acceleration = -g” and as the direction of the motion of object in free fall is vertically downward or upward so we can replace the “x 0″ with “y 0″ and “x” with “y” and get our final free fall acceleration equations as:

Equation 1:
Equation 2:
Equation 3:
Equation 4:
Equation 5:

Acceleration is said to be constant when the rate of change of velocity of an object is constant.

In many types of motion , The acceleration is either constant or approximately constant , For example when you drive a car and accelerate it , you accelerate it in almost constant rate.

When you are in a constant acceleration your Position-Time , Velocity-Time and Acceleration-Time curve looks like this:

Constant acceleration equations:

Constant acceleration or deceleration is so common in Physics and in life that a special set of equations are derived to analyze the situations in which acceleration is constant, Those equations are called “Constant acceleration equations”.

And these equations can only be applied to solve problems when the acceleration or deceleration is constant.

We can derive these equation using two approach one using simple approach and another using integral calculus.

First let us derive the equations using normal approach:

When the acceleration is constant , Both average acceleration and instantaneous acceleration are constant and equal so we can write the formula of constant acceleration and instantaneous acceleration as:

where , is the velocity at and is velocity at later time .

And we can re-write this equation as:

This equation is popularly known as first basic equation for constant acceleration.

Now We have,

The average velocity =

If  we replace the “v” in equation with the formula for “v” from first basic equation for constant acceleration which we derived above we get the following equation:

Now if we replace the with it’s formula and multiply both side of above equation by “t” then we get:

This equation is popularly known as second basic equation for constant acceleration.

Using these first and second basic equation for constant acceleration we can analyze almost every situation of constant acceleration , But we can also combine these two equation in different ways to get three more equations which are listed at the end of this page.

Now let us derive these equations using integral calculus:

We can write the derivative formula for acceleration in differential form as:

Integrating both side we get:

“a” is a constant so we can rewrite the equation as:

or,

or,

To evaluate the value of “c” we put “t=0″

Then at “t = 0″ , “v = v0″

So,

Thus we derived the first basic equation as:

Now let us derive the second basic equation for constant acceleration:

We can write the derivative formula for velocity in differential form as:

Now integrating both side we get:

Substituting “v” with the first basic equation:

Or,

As “v0″ and “a” are constant:

Now to evaluate “c” let us put “t=0″ and at “t=0″ “x = x0″

so

So ,

Constant acceleration equations list:

Equation 1:

Equation 2:

Equation 3:

Equation 4:

Equation 5:

When the velocity of a particle changes then it is said to undergo acceleration.

Acceleration is a vector quantity.

When velocity of the object increases then the acceleration is positive and when velocity decreases the acceleration is negative and is called deceleration.

Mathematically,

For an motion along an axis:

Average acceleration =

Where , “v2″ is the velocity at time “t2″ and “v1″ is the velocity at time “t1″.

And the limiting value of the average acceleration as tends towards zero , is called instantaneous acceleration or simply acceleration.

Mathematically,

Acceleration =

Or, Acceleration of a particle at a given constant is the rate of change of velocity at the instant. And acceleration is the derivative of  velocity of a particle and the second derivative of  it’s position( x ).

Mathematically:

Acceleration =

unit: The unit of acceleration in which it’s mathematical value is expressed is “meters per second squared” or , length per time squared.

Beside instantaneous velocity , average velocity is a way of defining how fast a body is moving.But  average velocity is only the average of the velocity of body over a given time interval ,  and not the exact measure of velocity of a body at an instant.

The exact measure of velocity of a body at an instant or over a very-very-very short time interval is known as instantaneous velocity.

Mathematically instantaneous velocity of an object is the limiting value  of average velocity as tends towards 0.

Or:

Instantaneous velocity =

And instantaneous velocity is also the derivative of the “position of the object” ( “x” ) with respect to “time interval”( “t” )

or, Instantaneous velocity =

In a graph of motion of an object drawn “x” versus “t”; Instantaneous velocity of the object is the slope of tangent line drawn at the position representing that instant.

Instantaneous velocity is a vector quantity and thus have  it’s direction ( positive or negative ) associated with it along with it’s magnitude.

Speed:

Speed of an object is the magnitude of the vector quantity “Instantaneous velocity” of the object.

Or , speed is similar to Instantaneous velocity and is different only because speed is a scalar quantity and doesn’t deals with the direction of the movement.

For example:

1. If the Instantaneous velocity of an object is  -7m/s then the speed of the object at that instance is 7m/s.

2. If the Instantaneous velocity of an object is  +19m/s then the speed of the object at that instance is 19m/s.

Where , = Total change in position , = total time interval.

= position of the object at two different time and = Different time.

Average velocity does not depend on the actual path followed by the object , It only depends on the initial and final position of the object in a time period.

If we draw a graph “x” versus “t”  for the motion of an object; where , “x” = position of an object and “t” = time.

Then the average velocity of the object in a given time interval is the slope of the line joining it’s initial and final position in the time interval.

Average Speed:

The average speed of a particle during a time interval is the average distance moved by the particle during the time interval.

Mathematically,

Average speed =

Example:

Let us suppose a vehicle which is moving in a strange way.

Let, at different time the vehicle is at different position from its starting point.

And let us also suppose the movement of vehicle in a direction as positive and in the opposite direction as negative, and the data for its position at different time is:

At time 0 second , it is at the origin.

At time 1 second, it is 1 meter far from the origin.

At time 2 second, it is 2 meter far from the origin.

At time 3 second, it is 7 meter far from the origin.

At time 4 second, it is 5 meter far from the origin.

At time 5 second, it is 10 meter far from the origin.

So we can analyze above data and come into following conclusion:

In the first second the vehicle moved 1 meter in positive direction.

In the second second the vehicle moved 1 meter in positive direction.

In the third second it moved 5 meters in positive direction.

In the fourth second it moved 2 meters in negative direction.

And in the last second it moved 5 meters in positive direction.

In the above example ,

the average velocity of the vehicle =

But the average speed of the vehicle =  

To do calculations and analysis in physics a standard is established for each of these three quantities.
Which are as follows:

Length:
The history of unit of length dates a long back. In A.D. 1120 the king of England decided that the standard of length in his country would be named the yard and would be precisely equal to the distance from the tip of his nose to the end of his outstretched arm. Similarly the unit feet was used in France. Many other systems for measuring length have been developed. As recently as 1960, the length of the meter was defined as the distance between two lines on a specific platinum–iridium bar stored under controlled conditions in France. This standard was abandoned for several reasons, a principal one being that the limited accuracy with which the separation between the lines on the bar can be determined does not meet the current requirements of science and technology. In the 1960s and 1970s, the meter was defined as 1 650 763.73 wavelengths of orange-red light emitted from a krypton-86 lamp. However, in October 1983, the meter (m) was redefined as the distance traveled by light in vacuum during a time of 1/299 792 458 second.

Mass:
The unit of mass Gram was originally derived from the mass of one centimeter cube of water. In 1887 The unit of mass Kilo Gram (KG) is defined as the mass of a specific platinum–iridium alloy cylinder kept at the International Bureau of Weights and Measures at Sevres, France. And since then the definition has not been changed since that time because platinum–iridium is an unusually stable alloy.

Time:
Before 1960, the standard of time was defined in terms of the mean solar day for the
year 1900. (A solar day is the time interval between successive appearances of the Sun
at the highest point it reaches in the sky each day.) The second was defined as of a mean solar day. The rotation of the Earth is now known to vary slightly with time, however, and therefore this motion is not a good one to use for defining a time standard. In 1967, the second was redefined to take advantage of the high precision attainable in a device known as an atomic clock, which uses the characteristic frequency of the cesium-133 atom as the “reference clock.” The second (s) is now defined as 9 192 631 770 times the period of vibration of radiation from the cesium atom.

Position of an object is its location. In science, to find the position of an object we take a reference point or already known location and compare the location of two. We regard a place as reference point as the origin and find the distance of the object from the origin to find it’s  location as shown in figure below.

in above Example there are two Objects whose position are x=-2m and x=3 m. Or the first one is 2m away from origin in negative direction and second one is 3 m away from origin in positive direction.

What is Displacement?

If an object is in motion then the position of the object will be changing with time.

In such case, The total change in of the position of the object is known as it’s Displacement. By formula:

Δx=x₂-x₁

or , The displacement of an object(Δx) can be calculated by subtracting it’s initial position(x₁) from it’s final position(x₂)

Note:

Displacement is a vector quantity or it has both magnitude and direction.

For example: If an object moves from position x=3 to x=7 then it’s displacement by formula is +4 (or 4)  or it moves 4 unit in positive direction. And if it moves from x=3 to x=0 then it’s displacement formula is  -3 unit or it moves 3 unit in negative direction.

Motion is the process of changing of position by an object. Everything around us is in motion even seemingly stationary things like buildings , road is moving along with rotation of earth , every planet orbits the sun which orbits center of galaxy and galaxy itself is in motion. The study and comparison of “Motion”  is called “Kinematics” in physics.

General properties of Motion:

a> The motion is always in a straight line. The line may be in any direction like horizontal , vertical etc. but it must be a straight line.

b> Motion is caused in an object due to the force (Push and Pull) applied on the object.

c> An object is considered to be in motion only if it moves like a single particle or every parts of the object is move in same direction and with same speed.