Relation between Velocity and Displacement:

When the acceleration of a particle is directly proportional to its displacement, we can establish a relationship between velocity and displacement by integrating the acceleration function. Let's analyze this relationship step by step:

1. Definition of terms:
- Acceleration (a): The rate of change of velocity with respect to time.
- Velocity (v): The rate of change of displacement with respect to time.
- Displacement (s): The change in position of a particle.

2. Given condition:
According to the given condition, the acceleration (a) is directly proportional to the displacement (s). Mathematically, we can express this as:
a ∝ s

3. Proportional constant:
Let's introduce a constant of proportionality (k) to relate acceleration and displacement. Therefore, we can write the equation as:
a = k * s

4. Relation between acceleration and velocity:
We know that acceleration is the rate of change of velocity. Thus, we can express this relationship as:
a = dv/dt

5. Differentiation:
To find the relation between velocity and displacement, we need to differentiate the equation a = k * s with respect to time (t). This gives us:
dv/dt = k * s

6. Integration:
Now, we integrate both sides of the equation with respect to displacement (s):
∫ dv = ∫ k * s ds

7. Integration results:
Integrating the left side with respect to velocity (v) and the right side with respect to displacement (s), we get:
v = k * (s^2)/2 + C

Here, C is the constant of integration.

8. Final equation:
Thus, the relationship between velocity (v) and displacement (s) can be expressed as:
v = k * (s^2)/2 + C

Summary:
When the acceleration of a particle is directly proportional to its displacement, the relationship between velocity and displacement can be expressed as v = k * (s^2)/2 + C, where k is the constant of proportionality and C is the constant of integration. This equation shows how the velocity of the particle changes as it undergoes displacement.

Let be a=kx
vdv/dx=kx
on integrated we get v=√k*x