?motion of a particle is governed by the equation s is equal to 2t^ - ...
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.