Question

derive equation of motion newton jaldi ​

Answer 1

Explanation:

There are mainly three equations of motion which describe the relationship between velocity, time, acceleration and displacement.

First, consider a body moving in a straight line with uniform acceleration. Then, let the initial velocity be u, acceleration is denoted as a, time period is denoted as t, velocity is denoted as v, and the distance travelled is denoted as S.

The equation of motions derivation can be done in three ways which are:

Derivation of equations of motion by Simple Algebraic Method

Derivation of Motion by Graphical Method

Derivation of Motion by Calculus Method

Below, the equations of motion are derived by all the three methods in a simple and easy to understand way.

Derivation of First Equation of Motion

The first equation of motion is:

v = u + at

Derivation of First Equation of Motion by Algebraic Method

It is known that the acceleration (a) of the body is defined as the rate of change of velocity.

So, the acceleration can be written as:

a = v − ut

From this, rearranging the terms, the first equation of motion is obtained, which is:

v = u + at

Derivation of First Equation of Motion by Graphical Method

Consider the diagram of the velocity-time graph of a body below:

Derivation Of Equation Of Motion

In this, the body is moving with an initial velocity of u at point A. The velocity of the body then changes from A to B in time t at a uniform rate. In the above diagram, BC is the final velocity, i.e. v after the body travels from A to B at a uniform acceleration of a. In the graph, OC is the time t. Then, a perpendicular is drawn from B to OC, a parallel line is drawn from A to D, and another perpendicular is drawn from B to OE (represented by dotted lines).

Following details are obtained from the graph above:

The initial velocity of the body, u = OA

The final velocity of the body, v = BC

From the graph,BC = BD + DC

So, v = BD + DC

v = BD + OA (since DC = OA)

Finally, v = BD + u (since OA = u) (Equation 1)

Now, since the slope of a velocity-time graph is equal to acceleration a,

So,

a = slope of line AB

a = BD/AD

Since AD = AC = t, the above equation becomes:

BD = at (Equation 2)

Now, combining Equation 1 & 2, the following is obtained:

v = at + u

Derivation of First Equation of Motion by Calculus Method

It is known that,

Derivation Of Equation Of Motion

So,

Derivation Of Equation Of Motion