Suppose the position of an object is given by ->r(vector) = (3.0t^2*ihat - 6.0t^3*jhat)m. Where t in seconds.
Determine its velocity ->v as a function of time t.
Determine its acceleration ->a as a function of time t.
Determine ->r at time t = 2.5 s.
Determine ->v at time t = 2.5s.
Express your answer using two significant figures. Express your answer in terms of the unit vectors ihat and jhat.
What it P representing? Position?
6t-18t
To determine the velocity ->v as a function of time t, we need to take the derivative of the position function ->r with respect to time t.
Given ->r(t) = (3.0t^2 * ihat - 6.0t^3 * jhat) m, we can differentiate each component with respect to t.
The derivative of (3.0t^2 * ihat) with respect to t is:
d/dt (3.0t^2 * ihat) = 6.0t * ihat
The derivative of (-6.0t^3 * jhat) with respect to t is:
d/dt (-6.0t^3 * jhat) = -18.0t^2 * jhat
So, the velocity ->v as a function of time t is:
->v(t) = 6.0t * ihat - 18.0t^2 * jhat
Next, to determine the acceleration ->a as a function of time t, we need to take the derivative of the velocity function ->v with respect to time t.
The derivative of (6.0t * ihat) with respect to t is:
d/dt (6.0t * ihat) = 6.0 * ihat
The derivative of (-18.0t^2 * jhat) with respect to t is:
d/dt (-18.0t^2 * jhat) = -36.0t * jhat
So, the acceleration ->a as a function of time t is:
->a(t) = 6.0 * ihat - 36.0t * jhat
To determine ->r at time t = 2.5s, substitute t = 2.5 into the position function ->r:
->r(2.5) = (3.0 * (2.5)^2 * ihat - 6.0 * (2.5)^3 * jhat) m
= (3.0 * 6.25 * ihat - 6.0 * 15.625 * jhat) m
= (18.75 * ihat - 93.75 * jhat) m
To determine ->v at time t = 2.5s, substitute t = 2.5 into the velocity function ->v:
->v(2.5) = (6.0 * 2.5 * ihat - 18.0 * (2.5)^2 * jhat)
= (6.0 * 2.5 * ihat - 18.0 * 6.25 * jhat)
= (15.0 * ihat - 112.5 * jhat)
So, ->r at time t = 2.5 s is (18.75 * ihat - 93.75 * jhat) m and ->v at time t = 2.5 s is (15.0 * ihat - 112.5 * jhat) m/s (expressed using two significant figures).
This is a problem of differentiation where the independent variable is t, and
r is the position vector.
v is the velocity vector.
a is the acceleration vector.
The motion is described in two orthogonal directions i and j, which means that you can do the calculations in each of the directions independently of each other.
Given
r = (3.0t^2i - 6.0t^3j) m
The i and j components of the position vector are
Pi(t)=3.0t^2
Pj(t)=6.0t^3
So d(Pi(t))/dt = d(3t²)/dt = 6t
...
can you complete the rest?
Post your answers for a check if you wish to.