Chapter 3 OutlineMotion in Two or Three Dimensions

• Position and velocity vectors

• Acceleration vectors

• Parallel and perpendicular components

• Projectile motion

• Uniform circular motion

• Relative velocity

Position Vector

• The position vector points from the origin to point , the position of the object.

• We can express the vector in terms of its , , and components.

• The position unit vector gives the direction from the origin to the object.

Velocity Vector

• The velocity vector is found from the time derivative of the position vector.

• The velocity is tangent to the path at each point.

• In component form:

Velocity Vector

• The velocity in each direction is just the time derivative of the coordinate of that direction.

• The magnitude of the velocity (speed) is given by:

Acceleration Vector

• The acceleration vector is found from the time derivative of the velocity vector.

• While we might typically think of acceleration as a change in speed, it is very important that we understand that it is a change in velocity.

• As we will discuss later in this chapter, in uniform circular motion, the speed is not changing, but the direction, and therefore velocity is constantly changing.

Acceleration Vector

• In component form:

• Or,

Parallel and PerpendicularComponents of Acceleration

• We can resolve the acceleration into its components parallel to the velocity (along the path) and perpendicular to the velocity.

• The parallel component, only changes the magnitude of the velocity, its speed.

• The perpendicular component, only changes the direction of the velocity, so its speed remains constant.

Projectile Motion

• Any body that is given an initial velocity and follows a path determined solely by the effects of gravity and air resistance is a projectile.

• The path the projectile follows is its trajectory.

• Initially, we will consider the simplest model in which we neglect the effects of air resistance, and the curvature of the earth.

Projectile Motion

• While a projectile moves in three-dimensional space, we can always reduce the problem to two dimensions by choosing to work in the vertical - plane that contains the initial velocity.

• We can simplify this further by treating the and components separately.

• The vertical and horizontal motions are independent

Projectile Motion

• In the ideal model, we only consider the force due to gravity, so there is no acceleration in the direction.

• In the direction, we have an acceleration due to gravity of downward. For the following equations, we will use a coordinate system in which up is positive.

• While not required, it is often simplest to set the origin as the initial position of the projectile, so that at .

Projectile Motion

• We can represent the initial velocity in terms of its components to rewrite the equations for position and velocity.

• We have also taken the initial position to be at the origin.

Trajectory Shape

• The previous equations tell us the position and velocity at each time, but to see the shape of the trajectory, we need to look at the vertical position as a function of horizontal position. ()

• Note that is a function of . This gives rise to a parabola.

Projectile Motion Example #1

Projectile Motion Example #2

Projectile Motion Exam Question Example

• Some nice pirates realize that their friends are carelessly sailing away without any cannonballs, so they decide to launch one towards the ship. The forgetful sailors are traveling directly away from the sailors at . The pirates’ cannon has a muzzle velocity of , and it is unfortunately stuck at an angle above the horizontal. If the pirates want to deliver the cannonball, they should fire the cannon when the ship is how far away?

Circular Motion

• Changes in direction mean changes in velocity.

• Special case: Uniform circular motion

• Constant speed

• No tangential acceleration, only perpendicular

Uniform Circular Motion

• Centripetal acceleration, • Constant magnitude

• Pointing towards center

• Period, • Time for one full circle

• Circumference () divided by speed

• Centripetal acceleration in terms of period

Nonuniform Circular Motion

• What if the speed is changing?

• Still have centripetal acceleration, , towards the center.

• Also have tangential acceleration, , along the path.

• Note that is the magnitude of the velocity.

• is not the same as !

Circular Motion Example

Relative Velocity

• When you are driving, and pass a car, it appears to be moving backwards.

• Relative to the ground, it is still moving forward.

• Relative to you, a moving frame of reference, it is moving backwards.

• What is the actual velocity?

• Any frame that is moving at a constant velocity is equally valid.

• This, along with the constant speed of light, is the basis of special relativity.

• Generally, if we do not explicitly state a frame, we measure relative to the ground.

Relative Position in One Dimension

• Notation:

• Object at position .

• In coordinate system , the object is at .

• In coordinate system , the object is at .

• The origin of coordinate system , with respect to , is at .

• Subscripts: object/frame

Relative Velocity in One Dimension

• Since velocity is the time derivative of position,

• If and are any two frames of reference,

Relative Velocity in Two or Three Dimensions

• We can extend this to two or three dimensions.

Relative Motion Example

Chapter 3 SummaryMotion in Two or Three Dimensions

• Position and velocity vectors

• Acceleration vector:

• Parallel component changes speed.

• Perpendicular component changes direction.

• Treat each component separately.

Chapter 3 SummaryMotion in Two or Three Dimensions

• Projectile motion• ;

• ;

• Circular motion

• Relative velocity

Chapter 4 OutlineNewton’s Laws of Motion

• Forces

• Contact and long range

• Superposition

• Newton’s first law

• Inertial frames of reference

• Newton’s second law

• Mass vs. weight

• Newton’s third law

• Inertial frames of reference

• Free-body diagrams

Forces

• Forces are interactions between two bodies or between a body and the environment.

• Contact force• Push, pull, friction…

• Long-range• Gravitational, electric, magnetic…

• Vectors!

Superposition of Forces

• When more than one force is acting on a body, the net force is the vector sum of the forces.

• Sometimes the net force is called the resultant force, .

Newton’s First Law

• Natural state of an object

• Aristotle, Galileo, Descartes…

• From Pricipia, the Latin followed by an English translation.

“Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.”

“Law I: Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.”

• From out text: A body acted on by no net force moves with constant velocity (which may be zero) and zero acceleration.