Constant acceleration & Constant acceleration equations





Constant acceleration:

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

 

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:

 a = a_{avg} = \dfrac{v - v_0}{t - 0}

where , v_0 is the velocity at t = 0 and v is velocity at later time t.

And we can re-write this equation as:

v = v_0 + at

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

Now We have,

The average velocity = v_{avg} = \frac{1}{2}(v_0 + v)

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:

v_{avg} = v_0 +\frac{1}{2} at

Now if we replace the v_{avg} with it’s formula \frac{x - x_0}{t - 0} and multiply both side of above equation by “t” then we get:

x - x_0 = v_0 t + \frac{1}{2} at^2

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:

dv = a. dt

Integrating both side we get:

\int dv =\int a. dt

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

or, \int dv = a. \int dt

or, v = at + c

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

Then at “t = 0″ , “v = v0″

So,

v_o = c

Thus we derived the first basic equation as:

v = at + v_0

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

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

dx = v. dt

Now integrating both side we get:

\int dx = \int v. dt

Substituting “v” with the first basic equation:

\int dx = \int (v_0 + at). dt

Or, \int dx = \int v_0 .dt + \int at. dt

As “v0″ and “a” are constant:

 x = v_0.t +\frac{1}{2} a t^2 +c

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

so x_0 = c

So ,  x - x_0 = v_0.t +\frac{1}{2} a t^2

Constant acceleration equations list:

Equation 1: v = v_0 + at

Equation 2:  x - x_0 = v_0.t +\frac{1}{2} a t^2

Equation 3: v^2 = v_0 ^2 + 2 a(x - x_0)

Equation 4:  x - x_0 = \frac{1}{2} (v_0 + v) t

Equation 5:  x - x_0 = vt - \frac{1}{2} at^2



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