The position vector r describes the path of an object moving in the xy-plane. Fi

Question

The position vector r describes the path of an object moving in the xy-plane. Find the velocity vector, speed, and acceleration vector the object. Evaluate the velocity vector and acceleration vector of the object at the given point. Sketch a graph of the path, and sketch the velocity and acceleration vectors at the given point. Position Vector Point r(t) = 3ti + (t - 1)j (3, 0) r(t) = ti + (-t^2 + 4)j (1, 3) r(t) = t^2i + tj (4, 2) r(t) = (1/4t^3 + 1)i + tj (3, 2) r(t) = 2 cos ti + 2 sin tj (Squareroot 2, Squareroot 2) r(t) = 3 cos ti + 2 sin tj (3, 0) r(t) = (t- sin t, 1 - cos t) (pi, 2)

Explanation / Answer

7)
a)
r(t) = <t-sin t , 1-cos t>

v(t) = r'(t)
= <1-cos t , sin t>

a(t) = v'(t)
= <sin t, cos t>

b)
at point (pi,2)
v(pi) =<1-cos pi , sin pi>
=<2, 0>

a(pi)= <sin pi , cos pi>
= <0, -1>

c)
v(t) = <1-cos t , sin t>
x = 1-cos t ---> 1-x = cos t
y = sin t

(1-x)^2 + y^2 = cos^2t + sin^2 t
= 1
(x-1)^2 + y^2 = 1
So, velocity curve is an unit radius circle centred at (1,0)

a(t) = <sin t, cos t>
x= sin t , y= cos t
x^2+y^2 = sin^2 t + cos ^2
=1
so,
x^2 + y^2 = 1
So, acceleration curve is an unit radius circle centred at origin

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