Position, Velocity, and Acceleration of a Particle in Rectilinear Motion

Given:
The position of a particle in rectilinear motion is defined by the relationship x=t^3-2t^2+10t-6, where x is in metres and t is in seconds.

To Find:
The particle's position, velocity, and acceleration at t=3 seconds.

Solution:
We are given the equation of the position function, x(t), which represents the position of the particle at any given time t. To find the position, velocity, and acceleration of the particle at t=3 seconds, we need to substitute t=3 into the position function.

1. Position:
The position of the particle at any given time t is given by the position function x(t). To find the position at t=3 seconds, we substitute t=3 into the function:
x(3) = (3)^3 - 2(3)^2 + 10(3) - 6
= 27 - 18 + 30 - 6
= 33 meters

Therefore, the particle's position at t=3 seconds is 33 meters.

2. Velocity:
The velocity of a particle is the derivative of its position function with respect to time. To find the velocity, we differentiate the position function, x(t), with respect to t.
v(t) = dx(t)/dt

Differentiating x(t) = t^3 - 2t^2 + 10t - 6 with respect to t:
v(t) = 3t^2 - 4t + 10

Substituting t=3 into the velocity function:
v(3) = 3(3)^2 - 4(3) + 10
= 27 - 12 + 10
= 25 m/s

Therefore, the particle's velocity at t=3 seconds is 25 m/s.

3. Acceleration:
The acceleration of a particle is the derivative of its velocity function with respect to time. To find the acceleration, we differentiate the velocity function, v(t), with respect to t.
a(t) = dv(t)/dt

Differentiating v(t) = 3t^2 - 4t + 10 with respect to t:
a(t) = 6t - 4

Substituting t=3 into the acceleration function:
a(3) = 6(3) - 4
= 14 m/s^2

Therefore, the particle's acceleration at t=3 seconds is 14 m/s^2.

Conclusion:
The position, velocity, and acceleration of the particle at t=3 seconds are as follows:
- Position: 33 meters
- Velocity: 25 m/s
- Acceleration: 14 m/s^2