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The equation r(t) = sin(4t)i + cos(4t)j​, 0t≥0 describes the motion of a particle moving along the unit circle. Answer the following questions about the behavior of the particle.

a. Does the particle have constant​ speed? If​ so, what is its constant​ speed?
b. Is the​ particle's acceleration vector always orthogonal to its velocity​ vector?
c. Does the particle move clockwise or counterclockwise around the​ circle?
d. Does the particle begin at the point (1,0)​?

User Jamiey
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Answer:

a) Particle has a constant speed of 4, b) Velocity and acceleration vector are orthogonal to each other, c) Clockwise, d) False, the particle begin at the point (0,1).

Explanation:

a) Let is find first the velocity vector by differentiation:


\vec v = (dr_(x))/(dt) i + \frac {dr_(y)}{dt} j


\vec v = 4\cdot \cos 4t\, i - 4 \cdot \sin 4t \,j


\vec v = 4 \cdot (\cos 4t \, i - \sin 4t\,j)

Where the resultant vector is the product of a unit vector and magnitude of the velocity vector (speed). Velocity vector has a constant speed only if magnitude of unit vector is constant in time. That is:


\|\vec u \| = 1

Then,


\| \vec u \| = \sqrt{\cos^(2) 4t + \sin^(2)4t }


\| \vec u \| = √(1)


\|\vec u \| = 1

Hence, the particle has a constant speed of 4.

b) The acceleration vector is obtained by deriving the velocity vector.


\vec a = (dv_(x))/(dt) i + \frac {dv_(y)}{dt} j


\vec a = 16\cdot (-\sin 4t \,i -\cos 4t \,j)

Velocity and acceleration are orthogonal to each other only if
\vec v \bullet \vec a = 0. Then,


\vec v \bullet \vec a = 64 \cdot (\cos 4t)\cdot (-\sin 4t) + 64 \cdot (-\sin 4t) \cdot (-\cos 4t)


\vec v \bullet \vec a = -64\cdot \sin 4t\cdot \cos 4t + 64 \cdot \sin 4t \cdot \cos 4t


\vec v \bullet \vec a = 0

Which demonstrates the orthogonality between velocity and acceleration vectors.

c) The particle is rotating clockwise as right-hand rule is applied to model vectors in 2 and 3 dimensions, which are associated with positive angles for position vector. That is:
t \geq 0

And cosine decrease and sine increase inasmuch as t becomes bigger.

d) Let evaluate the vector in
t = 0.


r(0) = \sin (4\cdot 0) \,i + \cos (4\cdot 0)\,j


r(0) = 0\,i + 1 \,j

False, the particle begin at the point (0,1).

User Omostan
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7.4k points
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