Acceleration in Terms of x

Given: t = √(x^2 - 1)

To find the acceleration in terms of x, we need to differentiate the equation for displacement with respect to time twice. This will give us the second derivative, which represents acceleration.

Differentiating t with respect to x:

dt/dx = (d/dx) √(x^2 - 1)
dt/dx = 1/2(x^2 - 1)^(-1/2) * (d/dx) (x^2 - 1)
dt/dx = x/√(x^2 - 1)

To find acceleration, we need to differentiate dt/dx with respect to x:

d^2t/dx^2 = (d/dx) (x/√(x^2 - 1))
d^2t/dx^2 = (1/√(x^2 - 1)) - x(1/2)(x^2 - 1)^(-3/2)(2x)
d^2t/dx^2 = 1/√(x^2 - 1) - x^2/√(x^2 - 1)(x^2 - 1)
d^2t/dx^2 = 1/√(x^2 - 1) - x^2/(x^2 - 1)

Simplifying the expression, we can combine the terms with a common denominator:

d^2t/dx^2 = (1 - x^2)/(x^2 - 1√(x^2 - 1))

Key Points:

- The equation t = √(x^2 - 1) relates the time and displacement of a particle along the positive x-axis.
- To find acceleration in terms of x, we differentiate the equation for displacement with respect to time twice.
- The first derivative dt/dx is obtained by differentiating the equation t with respect to x.
- The second derivative d^2t/dx^2 is obtained by differentiating the first derivative with respect to x.
- Simplifying the expression, we combine the terms with a common denominator and obtain the expression for acceleration in terms of x.

In conclusion, the acceleration in terms of x for the given equation t = √(x^2 - 1) is (1 - x^2)/(x^2 - 1√(x^2 - 1)).