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
velocity is the first derivative of position.
v=6.0t i -18.0 t^2 j
acceleration is the second derivative of position or
a= 6.0 i -36 t j
To determine the velocity ->v as a function of time t, we need to differentiate the position vector ->r with respect to time t.
Given ->r = (3.0t^2 * ihat - 6.0t^3 * jhat) m, we can differentiate each component with respect to time t.
Differentiating the x-component (3.0t^2 * ihat) with respect to time t gives us the x-component of the velocity ->v:
vx = d(3.0t^2)/dt = 6.0t
Differentiating the y-component (-6.0t^3 * jhat) with respect to time t gives us the y-component of the velocity ->v:
vy = d(-6.0t^3)/dt = -18.0t^2
Combining these components, the velocity vector ->v as a function of time t is:
->v = (6.0t * ihat - 18.0t^2 * jhat) m/s
To determine the acceleration ->a as a function of time t, we need to differentiate the velocity vector ->v with respect to time t.
Differentiating the x-component (6.0t * ihat) with respect to time t gives us the x-component of the acceleration ->a:
ax = d(6.0t)/dt = 6.0
Differentiating the y-component (-18.0t^2 * jhat) with respect to time t gives us the y-component of the acceleration ->a:
ay = d(-18.0t^2)/dt = -36.0t
Combining these components, the acceleration vector ->a as a function of time t is:
->a = (6.0 * ihat - 36.0t * jhat) m/s^2
To determine the position vector ->r at time t = 2.5 s, we substitute t = 2.5 into the given position vector ->r:
->r(2.5) = (3.0(2.5)^2 * ihat - 6.0(2.5)^3 * jhat) m
Calculating the expression gives us the position vector ->r at time t = 2.5 s.
To determine the velocity vector ->v at time t = 2.5 s, we substitute t = 2.5 into the velocity vector ->v:
->v(2.5) = (6.0(2.5) * ihat - 18.0(2.5)^2 * jhat) m/s
Calculating the expression gives us the velocity vector ->v at time t = 2.5 s.