Given:
The displacement of a point at time t is given by x² = 1/t².

To find:
The acceleration of the point at a given time t.

Solution:

To find the acceleration, we need to differentiate the displacement equation twice with respect to time (t).

1. Differentiating the displacement equation once:
Differentiating x² = 1/t² with respect to t, we get:

2x(dx/dt) = -2/t³

Simplifying the equation, we have:

2x(dx/dt) = -2/t³

Key Point:
The derivative of displacement with respect to time gives us the velocity, so dx/dt represents the velocity of the point at time t.

2. Substituting the given equation for x²:
We are given x² = 1/t², so we can substitute this equation into the previous equation:

2(1/t²)(dx/dt) = -2/t³

3. Differentiating the equation again:
Differentiating the equation obtained in step 2 with respect to t, we get:

2(1/t²)(d²x/dt²) + 2(2/t³)(dx/dt) = 6/t⁴

Simplifying the equation, we have:

(1/t²)(d²x/dt²) + (2/t³)(dx/dt) = 3/t⁴

Key Point:
The derivative of velocity with respect to time gives us the acceleration, so d²x/dt² represents the acceleration of the point at time t.

4. Substituting the given equation for x²:
Again, we substitute x² = 1/t² into the equation obtained in step 3:

(1/t²)(d²x/dt²) + (2/t³)(dx/dt) = 3/t⁴

5. Simplifying the equation:
Multiplying through by t⁴, we get:

(d²x/dt²)(t²) + 2(dx/dt) = 3

Key Point:
We have the acceleration equation in terms of displacement and velocity.

6. Substituting the value of dx/dt:
We know that dx/dt represents the velocity of the point at time t. We can substitute this value into the equation obtained in step 5:

(d²x/dt²)(t²) + 2(1/t²) = 3

Simplifying the equation, we have:

(d²x/dt²)(t²) + 2/t² = 3

7. Simplifying further:
To find the acceleration at a given time t, we can solve the equation obtained in step 6 for d²x/dt²:

(d²x/dt²)(t²) = 3 - 2/t²

Simplifying the right side of the equation, we have:

(d²x/dt²)(t²) = (3t² - 2)/t²

Key Point:
We have found the expression for acceleration in terms of time.

8.

x²=1+t²differentiate with respect to t2xdx/dt=2t ∴ xdx/dt=t velocity =dv/dt ∴ xd²x/dt²+(dx/dt)²=1 ∴(√1+t²)a+(t/x)²=1 ∴(√1+t²)a+t²/1+t²=1 ∴a(√1+t²)=1-t²/1+t² ∴a=(1-t²/1+t² )/(√1+t²) m/s²