?motion of a particle is governed by the equation s is equal to 2t^ - 3t^2 + 2t+2 , find the position ,velociand acceleration of the particle at time t is equal to 2seconds?

Question

Answer

Answer

Position, Velocity, and Acceleration of a Particle

Given Information

The motion of a particle is described by the equation:

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

The time at which we want to find the position, velocity, and acceleration is t = 2 seconds.

Finding Position

To find the position of the particle at t = 2 seconds, we simply plug in t = 2 into the equation:

s(2) = 2(2)^3 - 3(2)^2 + 2(2)^2 = 16 meters

So the position of the particle at t = 2 seconds is 16 meters.

Finding Velocity

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

v(t) = ds/dt = 6t^2 - 6t + 4

Now we can plug in t = 2 to find the velocity:

v(2) = 6(2)^2 - 6(2) + 4 = 16 meters/second

So the velocity of the particle at t = 2 seconds is 16 meters/second.

Finding Acceleration

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

a(t) = dv/dt = 12t - 6

Now we can plug in t = 2 to find the acceleration:

a(2) = 12(2) - 6 = 18 meters/second^2

So the acceleration of the particle at t = 2 seconds is 18 meters/second^2.

Conclusion

Therefore, the position of the particle at t = 2 seconds is 16 meters, the velocity is 16 meters/second, and the acceleration is 18 meters/second^2.

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