You CAN Ace Calculus

### 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

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

17calculus > integrals > linear motion

When using calculus for a useful application, the equations and subsequent derivatives usually mean something or describe something. On this page, we discuss the situation when a function represents the position of an object, in two dimension motion, vertically, horizontally or a combination. Beginning calculus topics are the only requirements to understand this material. If you've studied integration, even better. However, don't worry if you haven't. You can still understand most everything on this page. Just skip the parts with integration and come back here when you have. For a more advanced discussion, see the differential equations page on projectile motion.

Using Differentiation

Usually we derive or start with a function whose variable is time. This usually looks like $$s(t)$$ where the independent variable is t and represents time. The function itself is called a position function. This means that we can plug in a time, like $$t=3$$, to the function, $$s(3)$$ and the result is the ( instantaneous ) position of the object at time $$t=3$$. Of course the units are dependent upon what the equation represents. Also, we usually assume that time starts at $$t=0$$ and always increases.

When we have a position function, the first two derivatives have specific meanings. The first derivative is the velocity and the second derivative is the acceleration of the object. We take the derivative with respect to the independent variable, t.

The units of velocity are distance per unit time, in MKS units, meters per second, m/s.
The units of acceleration are distance per unit time squared, in MKS units, meters per second squared, m/s2.

Note About Velocity - - There is a difference between velocity and speed. Velocity can be negative and includes direction information. Speed is the magnitude of velocity and is always positive. Speed does not include direction. If you are familiar with vectors, velocity is a vector, speed is a scalar whose value is the magnitude of a velocity vector. Many people use these terms interchangable, which is incorrect.

Here are the equations you need.

Equations

position

$$s(t) = at^2/2 + v_0t + s_0$$

distance

velocity

$$s'(t) = v(t) = at + v_0$$

distance per unit time

acceleration

$$s''(t) = a(t) = a$$

distance per unit time squared

Note About Acceleration - - In the above table, we assumed acceleration is constant. This is true in many problems, especially ones where we talk about the acceleration due to gravity near the surface of the earth, for example. However, this is not true in ALL problems. So you need to pay attention to what is going on in the problem statement if you are required to set up these equations.

Constants and Variables

$$s_0$$

initial position; can also be written $$s(0)$$

$$v_0$$

initial velocity; can also be written $$v(0)$$

$$a$$

acceleration is often, but not always, constant

$$t$$

independent variable time

Okay, let's watch a video to get this into our heads a little bit more. Here is a good video that starts out by giving an overview of the derivative relationship between these equations. Then he does a quick example, and then finishes by giving a more intuitive description of the relationships. He does talk a little bit about anti-derivatives but if you aren't there yet, this video will still help you.

### PatrickJMT - Position, Velocity, Acceleration using Derivatives [8min-45secs]

video by PatrickJMT

Your best course of action now is to work as many practice problems as possible. These problems require only differentiation.

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. Here is a list converting the old numbering system to the new.

Linear Motion - Practice Problems Conversion

[1-685] - [2-686] - [3-689] - [4-690] - [5-691] - [6-692] - [7-693] - [8-1927] - [9-687]

[10-694] - [11-695] - [12-688]

Please update your notes to this new numbering system. The display of this conversion information is temporary.

GOT IT. THANKS!

Basic Problems

$$s(t)=3t^2+5t+2$$ describes a particle's motion with units in meters. Find a) the velocity and acceleration functions, and
b) the instantaneous velocity and acceleration at $$t=4$$ sec.

Problem Statement

$$s(t)=3t^2+5t+2$$ describes a particle's motion with units in meters. Find a) the velocity and acceleration functions, and
b) the instantaneous velocity and acceleration at $$t=4$$ sec.

a) $$v(t) = 6t+5~ m/s$$    $$a(t) = 6~ m/s^2$$
b) $$v(4)=29~ m/s$$    $$a(4) = 6~ m/s^2$$

Problem Statement

$$s(t)=3t^2+5t+2$$ describes a particle's motion with units in meters. Find a) the velocity and acceleration functions, and
b) the instantaneous velocity and acceleration at $$t=4$$ sec.

Solution

For this problem, we can tell that the position is in meters from the definition of $$s(t)$$ and we infer that time is in seconds from part b of the problem statement.
a) Since the velocity is the first derivative and the acceleration is the second derivative, we have the following.
$$v(t)=s'(t) = 6t+5$$
$$a(t)=v'(t)=s''(t) = 6$$
b) The instantaneous velocity and acceleration are given by letting $$t=4s$$ in the equations we found in part a.
$$v(4) = 6(4)+5 = 24+5=29$$
$$a(4) = 6$$

a) $$v(t) = 6t+5~ m/s$$    $$a(t) = 6~ m/s^2$$
b) $$v(4)=29~ m/s$$    $$a(4) = 6~ m/s^2$$

A particle moves along a path with its position $$p=t^2-6t+8$$ meters. a) When does the velocity equal 30 meters per second? b) When is the particle at rest?

Problem Statement

A particle moves along a path with its position $$p=t^2-6t+8$$ meters. a) When does the velocity equal 30 meters per second? b) When is the particle at rest?

Solution

He works this problem using the limit definition of the derivative, which is good practice but you can also work it by just taking the derivative. Of course, check with your instructor to see what they require.

### 686 solution video

video by PatrickJMT

If the position of a particle is given by $$x(t)=100-16t^2$$, find the position of the particle when the velocity is zero.

Problem Statement

If the position of a particle is given by $$x(t)=100-16t^2$$, find the position of the particle when the velocity is zero.

$$x = 100$$

Problem Statement

If the position of a particle is given by $$x(t)=100-16t^2$$, find the position of the particle when the velocity is zero.

Solution

### 689 solution video

video by Krista King Math

$$x = 100$$

A car with position function $$x(t)=100t-5t^2$$ is traveling $$100~ft/s$$ when the driver suddenly applies the brakes. How far and for how long does the car skid before it comes to a stop?

Problem Statement

A car with position function $$x(t)=100t-5t^2$$ is traveling $$100~ft/s$$ when the driver suddenly applies the brakes. How far and for how long does the car skid before it comes to a stop?

skids 500ft for 10secs

Problem Statement

A car with position function $$x(t)=100t-5t^2$$ is traveling $$100~ft/s$$ when the driver suddenly applies the brakes. How far and for how long does the car skid before it comes to a stop?

Solution

### 690 solution video

video by Krista King Math

skids 500ft for 10secs

The vertical position of a ball is given by $$y(t)=-16t^2+96t+50 ft$$. What is the maximum height the ball will reach?

Problem Statement

The vertical position of a ball is given by $$y(t)=-16t^2+96t+50 ft$$. What is the maximum height the ball will reach?

max height 194ft

Problem Statement

The vertical position of a ball is given by $$y(t)=-16t^2+96t+50 ft$$. What is the maximum height the ball will reach?

Solution

### 691 solution video

video by Krista King Math

max height 194ft

Intermediate Problems

A ball with position function $$y(t)=-gt^2/2+v_0t+s_0$$ is thrown straight upward from the ground with an initial velocity of 96 ft/sec. Find the maximum height that the ball attains and its velocity when it hits the ground.

Problem Statement

A ball with position function $$y(t)=-gt^2/2+v_0t+s_0$$ is thrown straight upward from the ground with an initial velocity of 96 ft/sec. Find the maximum height that the ball attains and its velocity when it hits the ground.

max height $$144~ft$$
velocity when it hits the ground $$-96~ft/sec$$

Problem Statement

A ball with position function $$y(t)=-gt^2/2+v_0t+s_0$$ is thrown straight upward from the ground with an initial velocity of 96 ft/sec. Find the maximum height that the ball attains and its velocity when it hits the ground.

Solution

### 692 solution video

video by Krista King Math

max height $$144~ft$$
velocity when it hits the ground $$-96~ft/sec$$

The motion of a ball is described by $$y(t)=-16t^2+64t$$. Find the maximum height and the velocity when the ball hits the ground.

Problem Statement

The motion of a ball is described by $$y(t)=-16t^2+64t$$. Find the maximum height and the velocity when the ball hits the ground.

max height $$64~f$$t
velocity $$-64~ft/sec$$

Problem Statement

The motion of a ball is described by $$y(t)=-16t^2+64t$$. Find the maximum height and the velocity when the ball hits the ground.

Solution

### 693 solution video

video by Krista King Math

max height $$64~f$$t
velocity $$-64~ft/sec$$

A coin is dropped from the roof of a 600ft tall building with initial velocity of -8 ft/sec. When does it hit the ground and what is its velocity at that point?

Problem Statement

A coin is dropped from the roof of a 600ft tall building with initial velocity of -8 ft/sec. When does it hit the ground and what is its velocity at that point?

Solution

### 1927 solution video

video by Krista King Math

Using Integration

Integration is required when you are given a velocity equation and you are asked to come up with the position function or when you are given acceleration and asked to come up with the velocity and/or the position function. So, essentially, it looks like this.

 derivative position → velocity → acceleration integral position ← velocity ← acceleration

Of course, with integration, an unknown constant is introduced. So, you will often be given a value that allows you to determine the unknown constant. This value is often called an initial condition. If the value is a position, it may be called an initial position. If it is a velocity, it may be called an intial velocity. You will find the word initial used a lot since most of the time, the value at time $$t=0$$ is given.

Let's work some practice problems to get an idea how this works.

A penny is thrown downward from a 300 foot tall tower with an initial velocity of 50 ft/sec. When does the penny hit the ground?

Problem Statement

A penny is thrown downward from a 300 foot tall tower with an initial velocity of 50 ft/sec. When does the penny hit the ground?

$$t=3.04\sec$$

Problem Statement

A penny is thrown downward from a 300 foot tall tower with an initial velocity of 50 ft/sec. When does the penny hit the ground?

Solution

### 687 solution video

video by PatrickJMT

$$t=3.04\sec$$

A particle has acceleration given by $$a(t)=12t-4$$ with initial velocity $$-10$$ and initial position $$0$$. Find its velocity and position functions.

Problem Statement

A particle has acceleration given by $$a(t)=12t-4$$ with initial velocity $$-10$$ and initial position $$0$$. Find its velocity and position functions.

$$v(t)=6t^2-4t-10$$
$$x(t)=2t^3-2t^2-10t$$

Problem Statement

A particle has acceleration given by $$a(t)=12t-4$$ with initial velocity $$-10$$ and initial position $$0$$. Find its velocity and position functions.

Solution

### 694 solution video

video by Krista King Math

$$v(t)=6t^2-4t-10$$
$$x(t)=2t^3-2t^2-10t$$

A ball is thrown straight upward from the ground with initial velocity 96 ft/s. How high does the ball rise and how long does it remain aloft?

Problem Statement

A ball is thrown straight upward from the ground with initial velocity 96 ft/s. How high does the ball rise and how long does it remain aloft?

Hint

If you are having trouble getting started on this problem, try using the equation $$v(t)=-gt+v_0$$ for velocity where g is the acceleration due to gravity. The negative sign indicates up as the positive direction.

Problem Statement

A ball is thrown straight upward from the ground with initial velocity 96 ft/s. How high does the ball rise and how long does it remain aloft?

max height $$144~ft$$; $$t=6~secs$$

Problem Statement

A ball is thrown straight upward from the ground with initial velocity 96 ft/s. How high does the ball rise and how long does it remain aloft?

Hint

If you are having trouble getting started on this problem, try using the equation $$v(t)=-gt+v_0$$ for velocity where g is the acceleration due to gravity. The negative sign indicates up as the positive direction.

Solution

### 695 solution video

video by Krista King Math

max height $$144~ft$$; $$t=6~secs$$

A car braked with a constant deceleration of 16 ft/s2, producing skid marks measuring 200ft before coming to a stop. How fast was the car traveling when the brakes were first applied?

Problem Statement

A car braked with a constant deceleration of 16 ft/s2, producing skid marks measuring 200ft before coming to a stop. How fast was the car traveling when the brakes were first applied?

$$v=80 ft/\sec$$

Problem Statement

A car braked with a constant deceleration of 16 ft/s2, producing skid marks measuring 200ft before coming to a stop. How fast was the car traveling when the brakes were first applied?

Solution

### 688 solution video

video by PatrickJMT

$$v=80 ft/\sec$$