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### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

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17calculus > vector functions > projectile motion

 Calculating The Velocity and Acceleration Calculating The Position Vector Acceleration Vector Components Practice

This page covers the basics of working with the position, velocity and acceleration vector functions. The acceleration vector that you learn about on this page can be expressed in terms of the unit tangent vector and the principal unit normal vector, which you can find on the acceleration vector components page.

Calculating The Velocity and Acceleration

Projectile motion using vector functions works just as you would expect. The following table lists the equations. If $$\vec{r}(t)$$ is a vector function describing the position of a projectile, the velocity is $$\vec{v}(t)=\vec{r}'(t)$$ and the acceleration is $$\vec{a}(t)=\vec{v}'(t)=\vec{r}''(t)$$.

Position $$\vec{r}(t) =x(t)\vhat{i}+y(t)\vhat{j}+z(t)\vhat{k}$$ $$\vec{v}(t)=x'(t)\vhat{i}+y'(t)\vhat{j}+z'(t)\vhat{k}$$ $$=\vec{r}'(t)$$ $$\| \vec{v}(t) \|$$ $$\vec{a}(t)=x''(t)\vhat{i}+y''(t)\vhat{j}+z''(t)\vhat{k}$$ $$=\vec{v}'(t)$$ $$=\vec{r}''(t)$$

There are really no surprises here. Notice that the speed is just the magnitude of the velocity and so it's value is always a positive scalar. Some instructors use the terms speed and velocity interchangeably but they actually refer to different things.

Calculating The Position Vector

Sometimes we are given the acceleration vector or the velocity vector and asked to calculate the position vector. In those cases we use integration. As you would expect, to get the velocity vector from the acceleration vector, we use these equations.

Acceleration $$\vec{a}(t)=a_x(t)\vhat{i}+a_y(t)\vhat{j}+a_z(t)\vhat{k}$$ $$\vec{v}(t)=\int{a_x(t)~dt}\vhat{i} + \int{a_y(t)~dt}\vhat{j} + \int{a_z(t)~dt}\vhat{k} + \vec{C} =$$ $$\int{\vec{a}(t)~dt} + \vec{C}$$

Note - - In the equation for the velocity vector, the vector $$\vec{C}$$ is the constant vector that we get when we do integration. The velocity vector could also be written $$\vec{v}(t)=\left[ \int{a_x(t)~dt} + C_x \right] \vhat{i} + \left[ \int{a_y(t)~dt} + C_y \right] \vhat{j} + \left[ \int{a_z(t)~dt} + C_z \right] \vhat{k}$$ where $$\vec{C} = C_x\vhat{i} + C_y\vhat{j} + C_z\vhat{k}$$.

Once we have the velocity vector (or if we are given the velocity vector), we can calculate the position vector. For the equations below, we assume the velocity vector is in the form $$\vec{v}(t) = v_x\vhat{i} + v_y\vhat{j} + v_z\vhat{k}$$.

Velocity $$\vec{v}(t) = v_x\vhat{i} + v_y\vhat{j} + v_z\vhat{k}$$ $$\vec{r}(t) = \int{v_x~dt}\vhat{i} + \int{v_y~dt}\vhat{j} + \int{v_z~dt}\vhat{k} + \vec{K}$$

In each of the above equations we end up with general constants, in our case $$\vec{C}$$ and $$\vec{K}$$. You will probably run across problems that give you information that you can use to find the actual values of these constants. Most of the time the information is given in the form of initial conditions, i.e. values of velocity and/or position at time $$t=0$$. However, values at any other time will also allow you to find the constants. To do this, you substitute the value for time into the final equation and evaluate. Some practice problems demonstrate how to do this.

Acceleration Vector Components

The acceleration vector $$\vec{a}(t)= x''(t)\vhat{i}+y''(t)\vhat{j}+z''(t)\vhat{k}$$ can be expressed in terms of two other unit vectors, the unit tangent vector and the principal unit normal vector. After working some practice problems, you need to learn how to calculate the other two unit vectors before learning how to write the acceleration using them.

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

Instructions - - Unless otherwise instructed, give your answers in exact form.

Find velocity and acceleration vectors when $$t = 0$$ for $$\vec{r}(t) = e^{2t}\hat{i} + e^{-t}\hat{j}$$

Problem Statement

Find velocity and acceleration vectors when $$t = 0$$ for $$\vec{r}(t) = e^{2t}\hat{i} + e^{-t}\hat{j}$$

 velocity $$\vec{v}(0) = \vec{r}'(0) = 2\hat{i} - \hat{j}$$ acceleration $$\vec{a}(0) = \vec{r}''(0) = 4\hat{i} + \hat{j}$$

Problem Statement

Find velocity and acceleration vectors when $$t = 0$$ for $$\vec{r}(t) = e^{2t}\hat{i} + e^{-t}\hat{j}$$

Solution

### 715 solution video

video by Krista King Math

 velocity $$\vec{v}(0) = \vec{r}'(0) = 2\hat{i} - \hat{j}$$ acceleration $$\vec{a}(0) = \vec{r}''(0) = 4\hat{i} + \hat{j}$$

Find velocity and acceleration at $$t=3/4$$ for $$\vec{r}(t) = 3\cos(2\pi t)\hat{i} +$$ $$3\sin(2\pi t)\hat{j}$$

Problem Statement

Find velocity and acceleration at $$t=3/4$$ for $$\vec{r}(t) = 3\cos(2\pi t)\hat{i} +$$ $$3\sin(2\pi t)\hat{j}$$

 velocity $$\vec{r}'(3/4) = 6\pi \hat{i}$$ acceleration $$\vec{r}''(3/4) = 12\pi^2\hat{j}$$

Problem Statement

Find velocity and acceleration at $$t=3/4$$ for $$\vec{r}(t) = 3\cos(2\pi t)\hat{i} +$$ $$3\sin(2\pi t)\hat{j}$$

Solution

### 716 solution video

video by Krista King Math

 velocity $$\vec{r}'(3/4) = 6\pi \hat{i}$$ acceleration $$\vec{r}''(3/4) = 12\pi^2\hat{j}$$

As time $$t>0$$ increases, a particle travels along the curve $$\mathcal{C}$$ with position parameterized by $$\vec{r}(t) = 2(\cos t+t\sin t)\hat{i} +$$ $$2(\sin t-t\cos t)\hat{j}$$. For each $$t>0$$, compute the tangent (velocity) vector and the speed. What is the length of the curve which the particle travels over between $$t=0$$ and $$t=4\pi$$?

Problem Statement

As time $$t>0$$ increases, a particle travels along the curve $$\mathcal{C}$$ with position parameterized by $$\vec{r}(t) = 2(\cos t+t\sin t)\hat{i} +$$ $$2(\sin t-t\cos t)\hat{j}$$. For each $$t>0$$, compute the tangent (velocity) vector and the speed. What is the length of the curve which the particle travels over between $$t=0$$ and $$t=4\pi$$?

Hint

For the length of the curve, use the equation $$L = \int_{a}^{b}{ \sqrt{x'(t)^2 + y'(t)^2 } ~dt }$$. See the vector functions arc length page for details.

Problem Statement

As time $$t>0$$ increases, a particle travels along the curve $$\mathcal{C}$$ with position parameterized by $$\vec{r}(t) = 2(\cos t+t\sin t)\hat{i} +$$ $$2(\sin t-t\cos t)\hat{j}$$. For each $$t>0$$, compute the tangent (velocity) vector and the speed. What is the length of the curve which the particle travels over between $$t=0$$ and $$t=4\pi$$?

Hint

For the length of the curve, use the equation $$L = \int_{a}^{b}{ \sqrt{x'(t)^2 + y'(t)^2 } ~dt }$$. See the vector functions arc length page for details.

Solution

In the middle of the solution for this problem is a part that you were not asked to do in the problem statement, from about 8 minutes to and 10 and a half minutes in the video.

### 706 solution video

video by Dr Chris Tisdell

Find velocity, acceleration and speed for the position function $$\vec{r}(t) = t\hat{i} + t^2\hat{j} + t^3\hat{k}$$.

Problem Statement

Find velocity, acceleration and speed for the position function $$\vec{r}(t) = t\hat{i} + t^2\hat{j} + t^3\hat{k}$$.

 velocity $$\vec{v}(t) = \vec{r}'(t) = \hat{i} + 2t\hat{j} + 3t^2\hat{k}$$ acceleration $$\vec{a}(t) = \vec{r}''(t) = 2\hat{j} + 6t\hat{k}$$ speed $$\| \vec{v}(t) \| = \sqrt{1+4t^2+9t^4}$$

Problem Statement

Find velocity, acceleration and speed for the position function $$\vec{r}(t) = t\hat{i} + t^2\hat{j} + t^3\hat{k}$$.

Solution

### 712 solution video

video by Krista King Math

 velocity $$\vec{v}(t) = \vec{r}'(t) = \hat{i} + 2t\hat{j} + 3t^2\hat{k}$$ acceleration $$\vec{a}(t) = \vec{r}''(t) = 2\hat{j} + 6t\hat{k}$$ speed $$\| \vec{v}(t) \| = \sqrt{1+4t^2+9t^4}$$

Find velocity, acceleration and speed for the position function $$\vec{r}(t) = t\hat{i}+3e^t\hat{j} + 4e^t\hat{k}$$.

Problem Statement

Find velocity, acceleration and speed for the position function $$\vec{r}(t) = t\hat{i}+3e^t\hat{j} + 4e^t\hat{k}$$.

 velocity $$\vec{v}(t) = \vec{r}'(t) = \hat{i} + 3e^t\hat{j} + 4e^t\hat{k}$$ acceleration $$\vec{a}(t) = \vec{r}''(t) = 3e^t\hat{j} + 4e^t\hat{k}$$ speed $$\| \vec{v}(t) \| = \sqrt{1+25e^{2t}}$$

Problem Statement

Find velocity, acceleration and speed for the position function $$\vec{r}(t) = t\hat{i}+3e^t\hat{j} + 4e^t\hat{k}$$.

Solution

### 713 solution video

video by Krista King Math

 velocity $$\vec{v}(t) = \vec{r}'(t) = \hat{i} + 3e^t\hat{j} + 4e^t\hat{k}$$ acceleration $$\vec{a}(t) = \vec{r}''(t) = 3e^t\hat{j} + 4e^t\hat{k}$$ speed $$\| \vec{v}(t) \| = \sqrt{1+25e^{2t}}$$

Find velocity, acceleration and speed for the position function $$\vec{r}(t) = (3\cos t)\hat{i} +$$ $$(3\sin t)\hat{j} - 4t\hat{k}$$.

Problem Statement

Find velocity, acceleration and speed for the position function $$\vec{r}(t) = (3\cos t)\hat{i} +$$ $$(3\sin t)\hat{j} - 4t\hat{k}$$.

 velocity $$\vec{v}(t) = \vec{r}'(t) = (-3\sin t)\hat{i} + (3\cos t)\hat{j} - 4\hat{k}$$ acceleration $$\vec{a}(t) = \vec{r}''(t) = (-3\cos t)\hat{i} - 3 \sin t \hat{j}$$ speed $$\| \vec{v}(t) \| = 5$$

Problem Statement

Find velocity, acceleration and speed for the position function $$\vec{r}(t) = (3\cos t)\hat{i} +$$ $$(3\sin t)\hat{j} - 4t\hat{k}$$.

Solution

### 714 solution video

video by Krista King Math

 velocity $$\vec{v}(t) = \vec{r}'(t) = (-3\sin t)\hat{i} + (3\cos t)\hat{j} - 4\hat{k}$$ acceleration $$\vec{a}(t) = \vec{r}''(t) = (-3\cos t)\hat{i} - 3 \sin t \hat{j}$$ speed $$\| \vec{v}(t) \| = 5$$

For the position function $$\vec{r}(t)=\langle t-\sin t,1-\cos t\rangle$$, find the velocity and acceleration vectors at the point $$(\pi,2)$$.

Problem Statement

For the position function $$\vec{r}(t)=\langle t-\sin t,1-\cos t\rangle$$, find the velocity and acceleration vectors at the point $$(\pi,2)$$.

$$\vec{v}(\pi) = \langle 2,0\rangle$$; $$\vec{a}(\pi) = \langle 0,-1 \rangle$$

Problem Statement

For the position function $$\vec{r}(t)=\langle t-\sin t,1-\cos t\rangle$$, find the velocity and acceleration vectors at the point $$(\pi,2)$$.

Solution

The equations we need are $$\vec{v}(t)=\vec{r}'(t)$$ and $$\vec{a}(t)=\vec{v}'(t)$$. So the velocity vector is $$\vec{v}(t)=\langle 1-\cos t, \sin t\rangle$$.
Now we need to find t. We are given the point $$(\pi,2)$$ on the position vector. So we have two equations we can use to find t,
$$t-\sin t=\pi$$ and $$1-\cos t=2$$. The second one is easier to solve for t, so we have
$$1-\cos t=2 \to$$ $$\cos t=-1 \to$$ $$t=\pi$$
Just to double-check, if we let $$t=\pi$$ in the equation $$t-\sin t=\pi$$, the answer checks.
So $$\vec{v}(\pi)=\langle 1-\cos \pi, \sin\pi\rangle$$ giving us $$\vec{v}(\pi)=\langle 2,0 \rangle$$.

To get the acceleration vector, we take the derivative of the velocity vector, i.e. $$\vec{a}(t)=\vec{v}'(t)$$. So $$\vec{a}(t)=\langle \sin t, \cos t\rangle$$ and at the point $$(\pi,2)$$ the acceleration vector is $$\vec{a}(\pi)=\langle 0,-1\rangle$$.
Although the problem statement did not ask for a graph, here is a graph of the position function, the velocity vector and the acceleration vector.

$$\vec{v}(\pi) = \langle 2,0\rangle$$; $$\vec{a}(\pi) = \langle 0,-1 \rangle$$

Find the position vector function $$\vec{r}(t)$$ for a particle with acceleration $$\vec{a}(t) = \langle 2t,2\sin(t),\cos(4t)\rangle$$, initial velocity $$\vec{v}(0)=\langle 1,-3,2\rangle$$ and initial position $$\vec{r}(0)=\langle 2,4,-1\rangle$$.

Problem Statement

Find the position vector function $$\vec{r}(t)$$ for a particle with acceleration $$\vec{a}(t) = \langle 2t,2\sin(t),\cos(4t)\rangle$$, initial velocity $$\vec{v}(0)=\langle 1,-3,2\rangle$$ and initial position $$\vec{r}(0)=\langle 2,4,-1\rangle$$.

$$\vec{r}(t) = \langle (1/3)t^3+t+2,$$ $$-2\sin(t)-t+4,$$ $$(-1/16)\cos(4t)+2t-15/16\rangle$$

Problem Statement

Find the position vector function $$\vec{r}(t)$$ for a particle with acceleration $$\vec{a}(t) = \langle 2t,2\sin(t),\cos(4t)\rangle$$, initial velocity $$\vec{v}(0)=\langle 1,-3,2\rangle$$ and initial position $$\vec{r}(0)=\langle 2,4,-1\rangle$$.

Solution

### 2056 solution video

video by MIP4U

$$\vec{r}(t) = \langle (1/3)t^3+t+2,$$ $$-2\sin(t)-t+4,$$ $$(-1/16)\cos(4t)+2t-15/16\rangle$$

Determine the velocity vector, speed and acceleration vector of an object when $$t=1$$ given by the position vector $$\vec{r}(t) = t\vhat{i} + (-0.5t^2+4)\vhat{j}$$.

Problem Statement

Determine the velocity vector, speed and acceleration vector of an object when $$t=1$$ given by the position vector $$\vec{r}(t) = t\vhat{i} + (-0.5t^2+4)\vhat{j}$$.

$$\vec{v}(1)=\langle 1,-1 \rangle$$, speed=$$\sqrt{2}$$, $$\vec{a}(1)=\langle 0,-1 \rangle$$

Problem Statement

Determine the velocity vector, speed and acceleration vector of an object when $$t=1$$ given by the position vector $$\vec{r}(t) = t\vhat{i} + (-0.5t^2+4)\vhat{j}$$.

Solution

### 2057 solution video

video by MIP4U

$$\vec{v}(1)=\langle 1,-1 \rangle$$, speed=$$\sqrt{2}$$, $$\vec{a}(1)=\langle 0,-1 \rangle$$

Determine the velocity vector, speed and acceleration vector of an object when $$t=2$$ for the position vector $$\vec{r}(t) = \cos(\pi t)\vhat{i} + \sin(\pi t)\vhat{j} +$$ $$(t^2/2)\vhat{k}$$

Problem Statement

Determine the velocity vector, speed and acceleration vector of an object when $$t=2$$ for the position vector $$\vec{r}(t) = \cos(\pi t)\vhat{i} + \sin(\pi t)\vhat{j} +$$ $$(t^2/2)\vhat{k}$$

$$\vec{v}(2) = \langle 0,\pi,2 \rangle$$, speed = $$\sqrt{\pi^2+4}$$, $$\vec{a}(2) = \langle -\pi^2,0,1 \rangle$$

Problem Statement

Determine the velocity vector, speed and acceleration vector of an object when $$t=2$$ for the position vector $$\vec{r}(t) = \cos(\pi t)\vhat{i} + \sin(\pi t)\vhat{j} +$$ $$(t^2/2)\vhat{k}$$

Solution

### 2058 solution video

video by MIP4U

$$\vec{v}(2) = \langle 0,\pi,2 \rangle$$, speed = $$\sqrt{\pi^2+4}$$, $$\vec{a}(2) = \langle -\pi^2,0,1 \rangle$$

Find the velocity and acceleration vectors and speed of the vector function $$\vec{r}(t) = \langle -5t,-3t^2,4t^4+3 \rangle$$ at $$t=1$$.

Problem Statement

Find the velocity and acceleration vectors and speed of the vector function $$\vec{r}(t) = \langle -5t,-3t^2,4t^4+3 \rangle$$ at $$t=1$$.

$$\vec{v}(t) = \langle -5,-6,16\rangle$$, $$\vec{a}(t) = \langle 0,-6,48 \rangle$$, speed = $$\sqrt{317}$$

Problem Statement

Find the velocity and acceleration vectors and speed of the vector function $$\vec{r}(t) = \langle -5t,-3t^2,4t^4+3 \rangle$$ at $$t=1$$.

Solution

### 2059 solution video

video by MIP4U

$$\vec{v}(t) = \langle -5,-6,16\rangle$$, $$\vec{a}(t) = \langle 0,-6,48 \rangle$$, speed = $$\sqrt{317}$$

A car travels with a velocity vector given by $$\vec{v}(t) = \langle t^2,e^t+1 \rangle$$, where t is measured in seconds and the vector components are measured in feet. If the initial position of the car is $$\vec{r}(0) = \langle 1,3 \rangle$$, find the position of the car after one second.

Problem Statement

A car travels with a velocity vector given by $$\vec{v}(t) = \langle t^2,e^t+1 \rangle$$, where t is measured in seconds and the vector components are measured in feet. If the initial position of the car is $$\vec{r}(0) = \langle 1,3 \rangle$$, find the position of the car after one second.

$$\vec{r}(1) = \langle 4/3, e+3 \rangle$$ feet

Problem Statement

A car travels with a velocity vector given by $$\vec{v}(t) = \langle t^2,e^t+1 \rangle$$, where t is measured in seconds and the vector components are measured in feet. If the initial position of the car is $$\vec{r}(0) = \langle 1,3 \rangle$$, find the position of the car after one second.

Solution

### 2060 solution video

video by PatrickJMT

$$\vec{r}(1) = \langle 4/3, e+3 \rangle$$ feet

A particle moves along a curve whose parametric equations are $$x(t) = e^{-t}$$, $$y(t) = 2\cos(3t)$$, $$z(t) = 2\sin(3t)$$. (a) Determine the velocity and acceleration vectors.
(b) Find the magnitude of the velocity (speed) and acceleration at $$t=0$$.

Problem Statement

A particle moves along a curve whose parametric equations are $$x(t) = e^{-t}$$, $$y(t) = 2\cos(3t)$$, $$z(t) = 2\sin(3t)$$. (a) Determine the velocity and acceleration vectors.
(b) Find the magnitude of the velocity (speed) and acceleration at $$t=0$$.

(a) $$\vec{v}(t) = -e^{-t}\vhat{i} - 6\sin(3t)\vhat{j} + 6\cos(3t)\vhat{k}$$; $$\vec{a}(t) = e^{-t}\vhat{i} - 18\cos(3t)\vhat{j} - 18\sin(3t)\vhat{k}$$
(b) $$\|\vec{v}(0)\| = \sqrt{37}$$; $$\|\vec{a}(0)\| = 5\sqrt{13}$$

Problem Statement

A particle moves along a curve whose parametric equations are $$x(t) = e^{-t}$$, $$y(t) = 2\cos(3t)$$, $$z(t) = 2\sin(3t)$$. (a) Determine the velocity and acceleration vectors.
(b) Find the magnitude of the velocity (speed) and acceleration at $$t=0$$.

Solution

### 2065 solution video

video by Dr Chris Tisdell

(a) $$\vec{v}(t) = -e^{-t}\vhat{i} - 6\sin(3t)\vhat{j} + 6\cos(3t)\vhat{k}$$; $$\vec{a}(t) = e^{-t}\vhat{i} - 18\cos(3t)\vhat{j} - 18\sin(3t)\vhat{k}$$
(b) $$\|\vec{v}(0)\| = \sqrt{37}$$; $$\|\vec{a}(0)\| = 5\sqrt{13}$$