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In Exercises 1 to 4, find the velocity and acceleration vectors and the equation of the tangent line for each of the following curves, at the given value of t.
\({\bf r}(t)=(\cos t){\bf i}+(\sin 2t){\bf j}, \,\hbox{at } t=0\)
\({\bf c}(t)=(t\,\sin t, t\,\cos t,\sqrt{3}t), \,\hbox{at } t=0\)
\({\bf r}(t)=\sqrt{2}t{\bf i}+e^t{\bf j}+e^{-t}{\bf k}, \,\hbox{at } t=0\)
\({\bf c}({t})=t{\bf i} + t {\bf j}+\frac{2}{3}t^{3/2}{\bf k}, \,\hbox{at } t=9\)
In Exercises 5 to 8, let \({\bf c}_1(t)=e^t{\bf i}+(\sin t){\bf j} + t^3{\bf k}\) and \({\bf c}_2(t)=e^{-t}{\bf i}+(\,\cos t){\bf j}-2t^3{\bf k}\). Find each of the stated derivatives in two different ways to verify the rules in the box preceding Example 1.
\(\displaystyle\frac{d}{{\it dt}}[{\bf c}_1(t)+{\bf c}_2(t)]\)
\(\displaystyle\frac{d}{{\it dt}}[{\bf c}_1(t)\,{\cdot}\, {\bf c}_2(t)]\)
\(\displaystyle\frac{d}{{\it dt}}[{\bf c}_1(t)\times {\bf c}_2(t)]\)
\(\displaystyle\frac{d}{{\it dt}}\{{\bf c}_1(t)\,{\cdot}\, [2{\bf c}_2(t)+{\bf c}_1(t)]\}\)
Consider the helix given by \(\textbf{c}(t)=(a\cos t, a\sin t, bt)\). Show that the acceleration vector is always parallel to the \(xy\) plane.
Prove the dot product rule.
Determine which of the following paths are regular:
Let \(\textbf{v}\) and \(\textbf{a}\) denote the velocity and acceleration vectors of a particle moving on a path \(\textbf{c}\). Suppose the initial position of the particle is \(\textbf{c}(0)=(3, 4, 0)\), the initial velocity is \(\textbf{v}(0)=(1, 1, -2)\), and the acceleration function is \(\textbf{a}(t)=(0, 0, 6)\). Find \(\textbf{v}(t)\) and \(\textbf{c}(t)\).
The acceleration, initial velocity, and initial position of a particle traveling through space are given by \[ \textbf{a}(t)=(2, -6, -4), \quad \textbf{v}(0)=(-5, 1, 3), \quad \textbf{r}(0)=(6, -2, 1). \] The particle’s trajectory intersects the \(yz\) plane exactly twice. Find these two intersection points.
The acceleration, initial velocity, and initial position of a particle traveling through space are given by \[ \textbf{a}(t)=(-6, 2, 4), \quad \textbf{v}(0)=(2, -5, 1), \quad \textbf{r}(0)=(-3, 6, 2). \] The particle’s trajectory intersects the \(yz\) plane exactly twice. Find these two intersection points.
If \({\bf r}(t)=6t{\bf i}+3t^2{\bf j}+t^3{\bf k}\), what force acts on a particle of mass \(m\) moving along \({\bf r}\) at \(t=0\)?
Let a particle of mass 1 gram (g) follow the path in Exercise 1, with units in seconds and centimeters. What force acts on it at \(t=0\)? (Give the units in your answer.)
A body of mass 2 kilograms moves on a circle of radius 3 meters, making one revolution every 5 seconds. Find the centripetal force acting on the body.
Find the centripetal force acting on a body of mass 4 kilograms (kg), moving on a circle of radius 10 meters (m) with a frequency of 2 revolutions per second (rps).
Show that if the acceleration of an object is always perpendicular to the velocity, then the speed of the object is constant. (HINT: See Example 1.)
Show that, at a local maximum or minimum of \(\|{\bf r}(t)\|\), the vector \({\bf r}'(t)\) is perpendicular to \({\bf r}(t)\).
A satellite is in a circular orbit 500 miles above the surface of the earth. What is the period of the orbit? (You may take the radius of the earth to be 4000 miles, or \({6.436\times10^6}\) meters.)
What is the acceleration of the satellite in Exercise 21? The centripetal force?
Find the path \({\bf c}\) such that \({\bf c}(0)=(0,-5,1)\) and \({\bf c}'(t)=(t,e^t,t^2)\).
Let \({\bf c}\) be a path in \({\mathbb R}^3\) with zero acceleration. Prove that \({\bf c}\) is a straight line or a point.
Find paths \({\bf c} (t)\) that represent the following curves or trajectories.
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Let \({\bf c}(t)\) be a path, \({\bf v}(t)\) its velocity, and \({\bf a}(t)\) the acceleration. Suppose \({\bf F}\) is a \(C^1\) mapping of \({\mathbb R}^3\) to \({\mathbb R}^3, m >0\), and \({\bf F}({\bf c}(t))=m{\bf a}(t)\) (Newton’s second law). Prove that \[ \frac{d}{{\it dt}}[m{\bf c}(t)\times {\bf v}(t)]={\bf c}(t)\times\, {\bf F}({\bf c}(t)) \] (i.e., “rate of change of angular momentum = torque”). What can you conclude if \({\bf F}({\bf c}(t))\) is parallel to \({\bf c}(t)\)? Is this the case in planetary motion?
Continue the investigations in Exercise 26 to prove Kepler’s law that a planet moving under the influence of gravity about the sun does so in a fixed plane.