Chapter 3

Given that \(\mathbf{r}(t)=\left\langle e^{-5 t} \sin t, e^{-5 t} \cos t, 4 e^{-5 t}\right\rangle\) is the position vector of a moving particle, find the following quantities: The speed of the particle

Given that \(\mathbf{r}(t)=\left\langle e^{-5 t} \sin t, e^{-5 t} \cos t, 4 e^{-5 t}\right\rangle\) is the position vector of a moving particle, find the following quantities: The acceleration of the particle

Find the maximum speed of a point on the circumference of an automobile tire of radius \(1 \mathrm{ft}\) when the automobile is traveling at \(55 \mathrm{mph}\).

A projectile is shot in the air from ground level with an initial velocity of \(500 \mathrm{~m} / \mathrm{sec}\) at an angle of \(60^{\circ}\) with the horizontal. The graph is shown here: . At what time does the projectile reach maximum height?

A projectile is shot in the air from ground level with an initial velocity of \(500 \mathrm{~m} / \mathrm{sec}\) at an angle of \(60^{\circ}\) with the horizontal. The graph is shown here: At what time is the maximum range of the projectile attained?

A projectile is shot in the air from ground level with an initial velocity of \(500 \mathrm{~m} / \mathrm{sec}\) at an angle of \(60^{\circ}\) with the horizontal. The graph is shown here: What is the total flight time of the projectile?

A projectile is fired at a height of \(1.5 \mathrm{~m}\) above the ground with an initial velocity of \(100 \mathrm{~m} / \mathrm{sec}\) and at an angle of \(30^{\circ}\) above the horizontal. Use this information to answer the following questions: Determine the maximum height of the projectile.

A projectile is fired at a height of \(1.5 \mathrm{~m}\) above the ground with an initial velocity of \(100 \mathrm{~m} / \mathrm{sec}\) and at an angle of \(30^{\circ}\) above the horizontal. Use this information to answer the following questions: Determine the range of the projectile.

A projectile is fired from ground level at an angle of \(8^{\circ}\) with the horizontal. The projectile is to have a range of \(50 \mathrm{~m}\). Find the minimum velocity necessary to achieve this range.

Prove that an object moving in a straight line at a constant speed has an acceleration of zero.

The acceleration of an object is given by \(\mathbf{a}(t)=t \mathbf{j}+t \mathbf{k} .\) The velocity at \(t=1 \sec\) is \(\mathbf{v}(1)=5 \mathbf{j}\) and the position of the object at \(t=1\) sec is \(\mathbf{r}(1)=0 \mathbf{i}+0 \mathbf{j}+0 \mathbf{k}\). Find the object's position at any time.

\(\begin{array}{ll}& \text { Find }\end{array}\) \(\mathbf{r}(t) \quad\) given \(\quad\) that \(\quad \mathbf{a}(t)=-32 \mathbf{j}\) \(\mathbf{v}(0)=600 \sqrt{3} \mathbf{i}+600 \mathbf{j},\) and \(\mathbf{r}(0)=\mathbf{0} .\)

Find the tangential and normal components of acceleration for \(\mathbf{r}(t)=a \cos (\omega t) \mathbf{i}+b \sin (\omega t) \mathbf{j}\) at \(t=0\).

Given \(\mathbf{r}(t)=t^{2} \mathbf{i}+2 t \mathbf{j}\) and \(t=1,\) find the tangential and normal components of acceleration.

Find the tangential and normal components of acceleration. \(\mathbf{r}(t)=\left\langle e^{t} \cos t, e^{t} \sin t, e^{t}\right\rangle\). The graph is shown here:

Find the tangential and normal components of acceleration. \(\mathbf{r}(t)=\langle\cos (2 t), \sin (2 t), 1\rangle\)

Find the tangential and normal components of acceleration. \(\mathbf{r}(t)=\left\langle 2 t, t^{2}, \frac{t^{3}}{3}\right\rangle\)

Find the tangential and normal components of acceleration. \(\mathbf{r}(t)=\left\langle\frac{2}{3}(1+t)^{3 / 2}, \frac{2}{3}(1-t)^{3 / 2}, \sqrt{2} t\right\rangle\)

Find the tangential and normal components of acceleration. \(\mathbf{r}(t)=\left\langle 6 t, 3 t^{2}, 2 t^{3}\right\rangle\)

Find the tangential and normal components of acceleration. \(\mathbf{r}(t)=t^{2} \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}\)

Find the tangential and normal components of acceleration. \(\mathbf{r}(t)=3 \cos (2 \pi t) \mathbf{i}+3 \sin (2 \pi t) \mathbf{j}\)

Find the position vector-valued function \(\mathbf{r}(t),\) given that \(\mathbf{a}(t)=\mathbf{i}+e^{t} \mathbf{j}, \quad \mathbf{v}(0)=2 \mathbf{j}, \quad\) and \(\mathbf{r}(0)=2 \mathbf{i}\)

An automobile that weighs \(2700 \mathrm{lb}\) makes a turn on a flat road while traveling at \(56 \mathrm{ft} / \mathrm{sec}\). If the radius of the turn is \(70 \mathrm{ft}\), what is the required frictional force to keep the car from skidding?

Find the time in years it takes the dwarf planet Pluto to make one orbit about the Sun given that \(a=39.5 \mathrm{~A} . \mathrm{U}\).

Suppose that the position function for an object in three dimensions is given by the equation \(\mathbf{r}(t)=t \cos (t) \mathbf{i}+t \sin (t) \mathbf{j}+3 t \mathbf{k}\) Show that the particle moves on a circular cone.

Suppose that the position function for an object in three dimensions is given by the equation \(\mathbf{r}(t)=t \cos (t) \mathbf{i}+t \sin (t) \mathbf{j}+3 t \mathbf{k}\) Find the angle between the velocity and acceleration vectors when \(t=1.5\).

Suppose that the position function for an object in three dimensions is given by the equation \(\mathbf{r}(t)=t \cos (t) \mathbf{i}+t \sin (t) \mathbf{j}+3 t \mathbf{k}\) Find the tangential and normal components of acceleration when \(t=1.5\).

True or False? Justify your answer with a proof or a counterexample. A parametric equation that passes through points \(\mathrm{P}\) and \(\mathrm{Q}\) can be given by \(\mathbf{r}(t)=\left\langle t^{2}, 3 t+1, t-2\right\rangle\), where \(P(1,4,-1)\) and \(Q(16,11,2)\).

True or False? Justify your answer with a proof or a counterexample. \(\quad \frac{d}{d t}[\mathbf{u}(t) \times \mathbf{u}(t)]=2 \mathbf{u}^{\prime}(t) \times \mathbf{u}(t)\)

True or False? Justify your answer with a proof or a counterexample. The curvature of a circle of radius \(r\) is constant everywhere. Furthermore, the curvature is equal to \(1 / r\).

Find the domains of the vector-valued functions. \(\quad \mathbf{r}(t)=\langle\sin (t), \ln (t), \sqrt{t}\rangle\)

Find the domains of the vector-valued functions. \(\quad \mathbf{r}(t)=\left\langle e^{t}, \frac{1}{\sqrt{4-t}}, \sec (t)\right\rangle\)

Sketch the curves for the following vector equations. Use a calculator if needed. \([\mathrm{T}] \mathrm{r}(t)=\left\langle t^{2}, t^{3}\right\rangle\)

Sketch the curves for the following vector equations. Use a calculator if needed. \(\mathbf{r}(t)=\left\langle\sin (20 t) e^{-t}, \cos (20 t) e^{-t}, e^{-t}\right\rangle\)

Find a vector function that describes the following curves. Intersection of the cylinder \(x^{2}+y^{2}=4\) with the plane \(x+z=6\)

Find a vector function that describes the following curves. Intersection of the cone \(z=\sqrt{x^{2}+y^{2}}\) and plane \(z=y-4\)

Find the derivatives of \(\mathbf{u}(t), \quad \mathbf{u}^{\prime}(t), \quad \mathbf{u}^{\prime}(t) \times \mathbf{u}(t),\) \(\mathbf{u}(t) \times \mathbf{u}^{\prime}(t),\) and \(\mathbf{u}(t) \cdot \mathbf{u}^{\prime}(t) .\) Find the unit tangent vector. \(\mathbf{u}(t)=\left\langle e^{t}, e^{-t}\right\rangle\)

Find the derivatives of \(\mathbf{u}(t), \quad \mathbf{u}^{\prime}(t), \quad \mathbf{u}^{\prime}(t) \times \mathbf{u}(t),\) \(\mathbf{u}(t) \times \mathbf{u}^{\prime}(t),\) and \(\mathbf{u}(t) \cdot \mathbf{u}^{\prime}(t) .\) Find the unit tangent vector. \(\mathbf{u}(t)=\left\langle t^{2}, 2 t+6,4 t^{5}-12\right\rangle\)

Evaluate the following integrals. \(\int\left(\tan (t) \sec (t) \mathbf{i}-t e^{3 t} \mathbf{j}\right) d t\)

Evaluate the following integrals. \(\int_{1}^{4} \mathbf{u}(t) d t,\) with \(\mathbf{u}(t)=\left\langle\frac{\ln (t)}{t}, \frac{1}{\sqrt{t}}, \sin \left(\frac{t \pi}{4}\right)\right\rangle\)

Find the length for the following curves. \(\mathbf{r}(t)=\langle 3(t), 4 \cos (t), 4 \sin (t)\rangle\) for \(1 \leq t \leq 4\)

Find the length for the following curves. \(\mathbf{r}(t)=2 \mathbf{i}+t \mathbf{j}+3 t^{2} \mathbf{k}\) for \(0 \leq t \leq 1\)

Reparameterize the following functions with respect to their are length measured from \(t=0\) in direction of increasing \(t\). \(\mathbf{r}(t)=\cos (2 t) \mathbf{i}+8 t \mathbf{j}-\sin (2 t) \mathbf{k}\)

Find the curvature for the following vector functions. \(\mathbf{r}(t)=(2 \sin t) \mathbf{i}-4 t \mathbf{j}+(2 \cos t) \mathbf{k}\)

Find the curvature for the following vector functions. \(\mathbf{r}(t)=\sqrt{2} e^{t} \mathbf{i}+\sqrt{2} e^{-t} \mathbf{j}+2 t \mathbf{k}\)

Find the unit tangent vector, the unit normal vector, and the \(\quad\) binormal \(\quad\) vector \(\quad\) for \(\mathbf{r}(t)=2 \cos t \mathbf{i}+3 t \mathbf{j}+2 \sin t \mathbf{k}\)

Find the tangential and normal acceleration components with the position vector \(\mathbf{r}(t)=\left\langle\cos t, \sin t, e^{t}\right\rangle .\)

A Ferris wheel car is moving at a constant speed \(v\) and has a constant radius \(r\). Find the tangential and normal acceleration of the Ferris wheel car.

The position of a particle is given by \(\mathbf{r}(t)=\left\langle t^{2}, \ln (t), \sin (\pi t)\right\rangle, \quad\) where \(t \quad\) is measured in seconds and \(\mathbf{r}\) is measured in meters. Find the velocity, acceleration, and speed functions. What are the position, velocity, speed, and acceleration of the particle at \(1 \mathrm{sec} ?\)

The following problems consider launching a cannonball out of a cannon. The cannonball is shot out of the cannon with an angle \(\theta\) and initial velocity \(\mathbf{v}_{0}\). The only force acting on the cannonball is gravity, so we begin with a constant acceleration \(\mathbf{a}(t)=-g \mathbf{j}\). . Find the velocity vector function \(\mathbf{v}(t)\).