Acceleration of Particle in Proportion to t^2
The equation X = (t^5)^-1 can be rewritten as X = t^-5, which means that X is inversely proportional to the fifth power of time. To find the acceleration of the particle, we need to differentiate X with respect to time twice.

• First Differentiation
dX/dt = -5t^-6
This gives us the velocity of the particle at any given time.

• Second Differentiation
d^2X/dt^2 = 30t^-7
This gives us the acceleration of the particle at any given time.

Therefore, the acceleration of the particle is proportional to t^-7.

Explanation
The negative exponent of t in the acceleration equation indicates that the acceleration decreases as time increases. This is because the distance travelled by the particle increases at a faster rate than the time taken to travel the distance.

The proportionality constant in the acceleration equation is 30, which means that the acceleration of the particle increases as the distance travelled decreases. This is because the particle has to cover a smaller distance in a shorter amount of time, resulting in a higher acceleration.

In conclusion, the acceleration of the particle is proportional to t^-7, which means that the acceleration decreases as time increases. The proportionality constant of 30 indicates that the acceleration increases as the distance travelled decreases.