Displacement and Velocity

The displacement of a particle moving along a straight line is given by the equation:

s = 3t^3 + 2t^2 - 10t + 9

To find the velocity at t = 2s, we need to take the derivative of the displacement equation with respect to time:

v = ds/dt = 9t^2 + 4t - 10

Plugging in t = 2s, we get:

v = 9(2)^2 + 4(2) - 10 = 40 m/s

Therefore, the velocity of the particle at t = 2s is 40 m/s.

Acceleration

To find the acceleration at t = 3s, we need to take the derivative of the velocity equation with respect to time:

a = dv/dt = 18t + 4

Plugging in t = 3s, we get:

a = 18(3) + 4 = 58 m/s^2

Therefore, the acceleration of the particle at t = 3s is 58 m/s^2.

Explanation

The displacement equation gives us the position of the particle at any given time. The velocity equation is the derivative of the displacement equation and gives us the rate of change of displacement with respect to time. The acceleration equation is the derivative of the velocity equation and gives us the rate of change of velocity with respect to time.

To find the velocity at t = 2s, we take the derivative of the displacement equation and plug in t = 2s. This gives us the velocity of the particle at that particular time.

To find the acceleration at t = 3s, we take the derivative of the velocity equation and plug in t = 3s. This gives us the acceleration of the particle at that particular time.