Question

how the variables of motion are related by differentiation and integration.

Answer 1

The variables of motion are

a(acceleration)

v(velocity)

s(displacement)

relation between in differential form is given by as below:-

and

Now relation between variables of motion in the integral form

Answer 2

Differentiation:

• If you have position (where something is), its derivative with respect to time gives velocity (how fast and in what direction it's moving).

• If you take the derivative of velocity, you get acceleration (how quickly the velocity is changing).

• 
  

1. Integration:

• If you have acceleration, integrating it over time gives you the velocity.

• Integrating velocity over time gives you back the position.

• 
  

Simpley :-

• Position is where you are.

• Velocity is the story of how you got there (your speed and direction over time).

• Acceleration is the "behind-the-scenes" force changing that story.

Differentiation peels back layers to see what’s driving motion, while integration builds up the bigger picture from those layers.

Example:

• If a car’s position is given by $s(t)=t^2$, then:

• Velocity $v(t)=\frac{ds}{dt}=2t$ (speed increases over time).

• Acceleration $a(t)=\frac{dv}{dt}=2$ (constant acceleration).

• 
  

Reverse it:

• Integrate acceleration $a(t)=2$ to get $v(t)=2t+C$.

• Integrate velocity to get back position $s(t)=t^2+Ct+D$.

It’s like a dance between cause and effect