Question

F = ma 2nd law of motion
derive this pls

Answer 1

Answer:

Velocity is defined as the rate of change of displacement. This is mathematically represented as:

$Velocity = \frac{Displacement}{Time}$

Rearranging, we get

Displacement = Velocity × Time

f the velocity is not constant then in the above equation we can use average velocity in the place of velocity and rewrite the equation as follows:

Displacement = $\frac{Initial Velocity + Final Velocity}{2}$ ×Time

Substituting the above equations with the notations used in the derivation of the first equation of motion, we get

s = $\frac{v+u}{2}$ × t

From the first equation of motion, we know that v = u + at. Putting this value of v in the above equation, we get

s =$\frac{u+(u+at)}{2}$ × t

s = $\frac{2u+at}{2}$ × t

s = $(\frac{2u}{2} +\frac{at}{2} )$ × t

s = $( u +\frac{1}{2} at)$  × t

On further simplification, the equation becomes:

$s = ut + \frac{1}{2} at^2$

Answer 2

Required solution:

According to the statement, let us assume that an object of mass m is moving in a straight line with an initial velocity u. Afterwards it is accelerated(uniformly) to the final velocity v and take time t. Now the constant force F is applied through the time t. The initial and final momentum of the object will be ${\sf{p_1}}$ as mu and the ${\sf{p_2}}$ as mv

Knowledge required: p = m × v i.e., Momentum = Mass × Velocity

The change in momentum

⠀⠀⠀⠀⠀⠀∝ ${\sf{p_2 \: - p_1}}$

⠀⠀⠀⠀⠀⠀∝ ${\sf{mv \: - mu}}$

⠀⠀⠀⠀⠀⠀∝ ${\sf{m(v-u)}}$

The rate of change of momentum

⠀⠀⠀⠀⠀⠀∝ ${\sf{\dfrac{m(v-u)}{t}}}$

Or when the force applied

⠀⠀⠀⠀⠀F ∝ ${\sf{\dfrac{m(v-u)}{t}}}$

⠀⠀⠀⠀⠀F = ${\sf{\dfrac{km(v-u)}{t}}}$

⠀⠀⠀⠀⠀⠀= kma

We write a at the place of ${\sf{\dfrac{(v-u)}{t}}}$ because of the formula of acceleration. Here, k is constant proportional.

⠀⠀⠀⠀⠀We already know that mass and acceleration have SI units as kg and m/s² respectively. So we can make the SI unit of force by using there two SI units.

⠀⠀⠀⠀⠀⠀⠀⠀⠀The unit of force is so chosen then k became one.

⠀⠀⠀⠀Knowledge required: The one unit of the force is defined as the the amount that produces an acceleration of 1 m/s² in an object of 1 kg mass. Therefore,

↪️ 1 unit of force = k(1 kg)(1 m/s²)

↪️ 1 unit of force = 1(1 kg)(1 m/s²)

↪️ F = kma

↪️ F = 1ma

↪️ F = ma Henceforth, derived!!