# Motion in Two Dimensions

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## Video Clips

#### Projectile Motion Example

RealVideo®
9:08 minutes (37:02 - 46:10)

Projectile motion equations derived from motion with constant acceleration; separating the x and y components.

Instructor: Prof. Walter Lewin
Prior Knowledge: 3D Motion (30:19 of V3), 1D Motion (34:18 of V2)

#### Projectile Motion Demonstration

RealVideo®
3:21 minutes (46:10 - 49:31)

Golf ball fired directly upward from moving cart and caught later by the cart.

Instructor: Prof. Walter Lewin
Prior Knowledge: Projectile Motion (37:02 of V3)

#### Projectile Motion: Shape and Maximum

RealVideo®
8:42 minutes (0:00 - 8:42)

Proof that PM is a parabola; calculation of maximum height for PM with qualitative explanation.

Prior Knowledge: Projectile Motion (37:02 of V3)

#### Horizontal Displacement for Projectile Motion

RealVideo®
5:06 minutes (8:42 - 13:48)

Calculation of horizontal displacement for PM and angle α that maximizes it.

Instructor: Prof. Walter Lewin
Prior Knowledge: Projectile Motion Shape (beginning of V4)

#### Horizontal Displacement Demo (1)

RealVideo®
11:24 minutes (13:48 - 25:12)

Demonstration of maximum horizontal displacement within calculated uncertainty for 45°.

Instructor: Prof. Walter Lewin
Prior Knowledge: Horizontal Displacement for Projectile Motion (8:42 of V4)

#### Horizontal Displacement Demo (2)

RealVideo®
10:18 minutes (25:12 - 35:30)

Demonstration of the same displacement for 30° and 60° with different times.

Instructor: Prof. Walter Lewin
Prior Knowledge: Horizontal Displacement for Projectile Motion (8:42 of V4)

#### Collisions in Free-fall

RealVideo®
6:25 minutes (35:30 - 41:55)

Example of monkey falling from a tree when gun is fired at it; proof that monkey gets hit independent of velocity.

Instructor: Prof. Walter Lewin
Prior Knowledge: Projectile Motion (37:02 of V3)

#### Free-fall Reference Frame

RealVideo®
3:28 minutes (41:55 - 45:23)

Proof that in the frame of reference of the monkey, the trajectory of bullet is straight, not an arc.

Instructor: Prof. Walter Lewin
Prior Knowledge: Collisions in Free-fall (35:30 of V4)

#### Monkey in Free-fall

RealVideo®
6:38 minutes (45:23 - 52:01)

Demonstration of ball hitting monkey in flight.

Instructor: Prof. Walter Lewin
Prior Knowledge: Free-fall Reference Frame (41:55 of V4)

#### Projectile Motion

RealVideo®
4:19 minutes (26:42 - 31:01)

Projectile equations of motion in x and y directions.

Instructor: Prof. Walter Lewin
Prior Knowledge: Projectile Motion (37:02 of V3)

#### Projectile Motion Example

RealVideo®
8:38 minutes (31:01 - 39:39)

Worked example of projectile motion for KC135 jet; x, y, x', y' determined and graph analyzed.

Instructor: Prof. Walter Lewin
Prior Knowledge: Projectile Motion (37:02 of V3)

## Lecture Notes

#### Vectors

PDF - 1.6 MB#
Page 1

Definition of a vector, displacement.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None

#### Vector Operations

PDF - 1.6 MB#
Page 1 to page 2

Addition, subtraction, negative of a vector; multiplying scalars and vectors.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Vectors

#### Vector Components

PDF - 1.6 MB#
Page 3 to page 4

Components of vectors in two or three dimensions, with examples; unit vectors; arbitrary vector in terms of unit vectors example.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Vector Operations

#### Vector Multiplication

PDF - 1.6 MB#
Page 5 to page 6

Definition of dot product, with example; distributive law.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Vector Components

#### Cross Products

PDF - 1.4 MB#
Page 1 to page 2

Definition of cross product of two vectors; right-hand rule; 2nd and 3rd order determinants; cross product examples.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 4

#### Motion in Three Dimensions

PDF - 1.4 MB#
Page 2 to page 5

Extension of one-dimensional motion to three dimensions. Includes average and instantaneous velocity, average and instantaneous acceleration, and constant acceleration.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lectures 3 and 4

#### Projectile Motion

PDF - 1.3 MB#
Page 1 to page 4

Equations for ballistic motion in two dimensions. Includes examples.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 5

#### Projectile Problem - Solving Strategy

PDF - 1.3 MB#
Page 5

List of steps for solving projectile motion problems in two-dimensions. Includes example.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Projectile Motion

#### Circular Motion

PDF - 1.3 MB#
Page 1 to page 5

Uniform circular motion; centripetal acceleration; non-uniform circular motion; reference frames; relativity of motion; problem-solving strategy for relative velocities with examples.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 6

#### Coordinate Systems

PDF#
Page 6 to page 9

Overview of rectangular, spherical, and cylindrical coordinate systems.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None

#### Vectors

PDF#
Page 1 to page 3

Definition of a vector, displacement.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None

#### Vector Operations

PDF#
Page 4 to page 5

Addition, subtraction, negative of a vector; multiplying scalars and vectors.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Vectors

#### Vector Components

PDF#
Page 6 to page 11

Components of vectors in two or three dimensions, with examples; unit vectors; arbitrary vector in terms of unit vectors example.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Vector Operations

#### Vector Multiplication

PDF#
Page 12 to page 15

Definition of dot product, with example; distributive law.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Vector Components

#### Cross Products

PDF#
Page 1 to page 6

Definition for two vectors; right-hand rule; 2nd and 3rd order determinants; cross product examples.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 4

#### Motion in Three Dimensions

PDF#
Page 7 to page 18

Extension of one-dimensional motion to three dimensions. Includes average and instantaneous velocity, average and instantaneous acceleration, and constant acceleration.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lectures 3 and 4

#### Projectile Motion

PDF#
Page 1 to page 11

Equations for ballistic motion in two dimensions. Includes examples.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 5

#### Projectile Problem-Solving Strategy

PDF#
Page 12 to page 14

List of steps for solving projectile motion problems in two-dimensions. Includes example.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Projectile Motion

#### Circular Motion

PDF#
Page 1 to page 14

Uniform circular motion; centripetal acceleration; non-uniform circular motion; reference frames; relativity of motion; problem-solving strategy for relative velocities with examples.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 6

#### Vector Example Problem

PDF
Page 1 to page 2

Performing vector addition using vector components; vector component notation.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Multi-Dimensional Kinematics

PDF
Page 1 to page 2

Velocity and acceleration in multiple dimensions; special case of projectile motion; solving quadratic equations.

Instructor: Dr. George Stephans
Prior Knowledge: 1D Kinematics

#### Projectile and Circular Motion

PDF
Page 1 to page 2

Projectile motion equations; circular motion equations; centripetal acceleration, with equation; dimensional analysis method.

Instructor: Dr. George Stephans
Prior Knowledge: Multi-Dimensional Kinematics

#### Two Dimensional Motion

PDF#
Page 1 to page 10

Vector description of motion, including equations for position, velocity and acceleration; projectile motion; kinematic equations of motion in terms of x- and y- components; initial conditions; orbit equation with derivation.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: Lecture 2

#### Experiment 2

PDF#
Page 11 to page 17

Projectile motion experiment set-up and procedure.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: 2D Motion

## Practice Problems

#### Indoor Cannon

PDF
Problem 7

3-part problem; maximizing distance for cannon to shoot when constrained by ceiling.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None

PDF
Problem 1 to problem 2

Determining what conditions must apply for force vectors to add up to zero. Solution not included.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Kinematics in Two Dimensions

PDF
Problem 1 to problem 1

Velocity and acceleration of projectile motion. Solution not included.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Throwing a Stone Down a Hill

PDF
Problem 2

Finding the landing point of a projectile on an inclined plane.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Hitting the Bucket

PDF
Problem 3

Hitting a falling body with a projectile.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Law of Addition of Velocities

PDF
Problem 7

Velocities in relatively inertial reference frames.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Gravitational Slingshot

PDF
Problem 8

Acceleration in relatively inertial reference frames.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Roller-coaster

PDF
Problem 8

Speed and acceleration of a cart on a roller-coaster. Solution not included.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Two Projectiles

PDF
Problem 1

Projectile motion of two objects. Solution not included.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Parabolic Orbit

PDF
Problem 2

Magnitude and direction of velocity for a parabolic orbit. Solution not included.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Projectile and Falling Person

PDF
Problem 3

Motion of a stone thrown at a falling person. Solution not included.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Projectile Motion

PDF
Problem 4

Speed and acceleration at the apex of projectile's path. Solution not included.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Projectile Motion

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
PDF
Problem 3

Motion of a fielder running to catch a softball.

Prior Knowledge: None

#### Second Law and Projectile Motion

PDF
Problem 4

Motion of an object sliding off an inclined roof and falling to the ground.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Triangle Inequality

PDF#
Problem 6

Finding maximum distance between two towns.

Instructor: Prof. Walter Lewin
Prior Knowledge: None

#### Orthogonality

PDF#
Problem 7

Finding a parameter that makes two vectors orthogonal.

Instructor: Prof. Walter Lewin
Prior Knowledge: None

#### Perpendicular Unit Vectors

PDF#
Problem 9

Finding two unit vectors perpendicular to two given vectors.

Instructor: Prof. Walter Lewin
Prior Knowledge: None

#### 3D Kinematics

PDF#
Problem 10

3-part 3D kinematics problem; finding v, |v|, a.

Instructor: Prof. Walter Lewin
Prior Knowledge: None

## Exam Questions

#### Surveillance Balloon

PDF#
Problem 2

3-part projectile problem; finding time and height of bullet hitting balloon, and time before firing (at fixed v, θ). Solutions are given at the end of each problem.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None

#### Crossing a River

PDF#
Problem 3

For boat crossing a flowing river, finding angle for direct crossing and for optimal time. Solutions are given at the end of each problem.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None

#### Two Walkers

PDF#
Problem 2

3-part 2D kinematics; finding point where walkers on square meet, distance to (0,0), and average velocity. Solutions are given at the end of each problem.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None

#### Bomb Targeting

PDF#
Problem 3

4-part problem; finding time and distance to impact, speed and location of impact. Solutions are given at the end of each problem.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None

#### Projectile Motion

PDF
Problem 2

Motion of a rock thrown upward from a bridge.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Reference Frames

PDF
Problem 3

Motion of a rock thrown from a bridge as seen from multiple reference frames.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Velocity and Acceleration

PDF
Problem 7

Finding and plotting the velocity and acceleration of a particle given equations for its position.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Ball and Boat

PDF
Problem 10

Motion of a ball thrown from a moving boat.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Collision of Two Projectiles

PDF
Problem 11

Motion of an acrobat catching a ball in mid-air.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Different Punt Angles

PDF
Problem 2

Comparing distance and hangtime for a punt at two different angles.

Instructors: Dr. Peter Dourmashkin, Prof. Kate Scholberg
Prior Knowledge: None

#### Second Law and Projectile Motion

PDF
Problem 2

Motion of an object sliding off an inclined roof and falling to the ground.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

PDF
Problem 4

Motion of a boat rowing across a river perpendicular to the current.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Projectile Motion

PDF
Problem 6

Motion of a ball projected upwards and at an angle.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Projectile Motion

PDF
Problem B8

Flight of a baseball.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Reference Frames

PDF#
Problem 1a

Motion of a thrown ball as seen by observers at different reference frames.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Projectile Motion

PDF#
Problem 3

Motion of an object that slides off the top step of a staircase.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Projectile Motion

PDF#
Problem 1

3-part projectile motion problem involving stones thrown into the air.

Instructor: Prof. Walter Lewin
Prior Knowledge: None

#### 3D and 1D Kinematics

PDF#
Problem 2

3-part 3D kinematics and 2-part 1D kinematics problem about a moving particle.

Instructor: Prof. Walter Lewin
Prior Knowledge: None