if displacement of a particle is directly proportional to square root of time the acceleration is proportional to

If the displacement (\( s \)) of a particle is directly proportional to the square root of time (\( t \)), we can write this relationship as:
\[ s \propto \sqrt{t} \]
or
\[ s = k \sqrt{t} \]
where \( k \) is a constant of proportionality.

To find the relationship between acceleration and time, we need to derive the expressions for velocity and acceleration.

1. **Velocity** (\( v \)) is the first derivative of displacement with respect to time:
\[ v = \frac{ds}{dt} \]

Given \( s = k \sqrt{t} \), we find:
\[ v = \frac{d}{dt}(k \sqrt{t}) = k \cdot \frac{d}{dt}(\sqrt{t}) = k \cdot \frac{1}{2\sqrt{t}} = \frac{k}{2\sqrt{t}} \]

2. **Acceleration** (\( a \)) is the first derivative of velocity with respect to time:
\[ a = \frac{dv}{dt} \]

Given \( v = \frac{k}{2\sqrt{t}} \), we find:
\[ a = \frac{d}{dt} \left( \frac{k}{2\sqrt{t}} \right) = \frac{k}{2} \cdot \frac{d}{dt} (t^{-\frac{1}{2}}) = \frac{k}{2} \cdot \left( -\frac{1}{2} t^{-\frac{3}{2}} \right) = -\frac{k}{4} t^{-\frac{3}{2}} \]

Thus, the acceleration \( a \) is proportional to \( t^{-\frac{3}{2}} \).

In summary, the acceleration of the particle is inversely proportional to the cube of the square root of time, which can be expressed as:
\[ a \propto t^{-\frac{3}{2}} \]

Would you like more details or have any questions?

Here are 5 questions you might consider next:
1. How do you derive the second derivative of a function?
2. What is the physical significance of acceleration being inversely proportional to time raised to a power?
3. Can you explain the relationship between displacement, velocity, and acceleration in more detail?
4. How does this example apply to real-world scenarios?
5. What are some other examples where displacement is proportional to different functions of time?

**Tip:** Remember, understanding how to take derivatives and interpret their physical meaning is crucial in kinematics and dynamics.

Explore how acceleration relates to displacement and time in proportional motion. Learn about derivatives, kinematics, and the relationship between acceleration and time. Suitable for high school students studying physics and calculus.

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