Question

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)​?

Answer 1

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).

Step-by-step explanation:

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

$\vec v = \frac{dr_{x}}{dt} i + \frac {dr_{y}}{dt} j$

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

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

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,

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

$\| \vec u \| = \sqrt{1}$

$\|\vec u \| = 1$

Hence, the particle has a constant speed of 4.

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

$\vec a = \frac{dv_{x}}{dt} i + \frac {dv_{y}}{dt} j$

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

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).