Homework 5: Moving Frames and Acceleration

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SOME PROBLEMS COPIED FROM HANDOUTS - NOT SURE OF THEIR SOURCE

Completion Packet

Find the velocity, acceleration and speed of a particle with the position function $\mathbf {x}(t) = (t, t^{2}, 2)$ . Sketch the path of the particle by hand, and draw the velocity and acceleration vectors on your graph for $t=1$ .

Find the velocity, acceleration and speed of a particle with the position function $\mathbf {x}(t) = (t, 2 \cos {t}, \sin {t})$ . Sketch the path of the particle by hand, and draw the velocity and acceleration vectors on your graph for $t=0$ .

Find the moving frame for the path $\vec {x}(t) = (2t, 6\sin (3t), -6\cos (3t))$ .

Find the the acceleration of the path $\vec {x}(t) = (t,t^2)$ , and express it as a linear combination of the unit tangent vector and the unit normal vector.

Find the the acceleration of the path $\vec {x}(t) = (e^t, t)$ , and express it as a linear combination of the unit tangent vector and the unit normal vector.

Suppose we have a path $\vec {x}(t)$ and a point $\vec {c}$ in $\mathbb {R}^3$ such that $\|\vec {x}(t)-\vec {c}\|=R$ for some constant $R$ and for all $t$ .

(a): Geometrically, what can you say about this path?
(b): Prove that $\vec {v}(t)$ is perpendicular to $\vec {x}(t)-\vec {c}$ .
(c): If $\vec {x}(t)$ has constant speed, show that $\kappa (t)$ is nonzero for all $t$ .

Prove that if a particle moves with constant speed, then the velocity and acceleration vectors are orthogonal.

Find the tangential and normal components of the acceleration vector for $\mathbf {x}(t)=(1+t,t^{2}-2t)$ .

Find the tangential and normal components of the acceleration vector for $\mathbf {x}(t)=(t, t^{2}, 3t)$ .

Compute the following limits. Show all work. (“Wolfram Alpha” is not work.)

(a): $\displaystyle \lim _{x\to 3} \cfrac {x^2-9}{x^2+2x-3}$
(b): $\displaystyle \lim _{x\to \pi ^-} \ln (\sin {x})$
(c): $\displaystyle \lim _{v\to 4^+} \frac {4-v}{|4-v|}$
(d): $\displaystyle \lim _{x\to \infty } (\sqrt {x^2+4x+1} - x)$
(e): $\displaystyle \lim _{x\to \infty } e^{x-x^2}$
(f): $\displaystyle \lim _{x\to 0} \frac {\sin {2x}}{x}$

Graded Problems

Find the moving frame for the path $\vec {x}(t) = (e^t\cos (t), e^t\sin (t), 5)$ .

Prove that $\frac {|(\vec {v}\times \vec {a})\cdot \vec {a}'|}{\|\vec {v}\times \vec {a}\|^2} = \frac {\left \|d\vec {B}/dt\right \|}{|ds/dt|}.$

Professional Problem

(a): Beginning with the formula $\mathbf {a}=s'' \mathbf {T} + (s')^{2}\kappa \mathbf {N}$ , cross both sides with the velocity vector $\mathbf {v}=s'\mathbf {T}$ and derive the formula $\kappa = \frac {||\mathbf {v} \times \mathbf {a}||}{(s')^{3}}.$ Make sure to write your solution in your own words!
(b): Let $y=f(x)$ . The graph of $y=f(x)$ is therefore a plane curve. Assuming $f \in C^2$ (that is, it is twice differentiable and its second derivatives are continuous), use the formula from (a) to prove the following well-known formula for the curvature of such a curve: $\kappa (x) = \dfrac {|f''(x)|}{ \left [ 1 + (f'(x))^2 \right ]^{3/2}}.$