Notes:Color Guide Gold : Important concept. Write this down. Orange : Definition. Write this down. Blue : Important information, but you do not need to copy. Red : Example. Copy if needed. White : Will be discussed by Ms. King.

Chapter 2: Motion in One Dimension Section 1: Displacement & Velocity

Definition: motion – the change of an object’s position relative to some reference point. In order to distinguish motion, a frame of reference must be used. The signal light, power line poles, and hills can be used as a frame of reference.

Objects that are “at rest” are not in motion. Definition: displacement (∆x) – the change in position of an object.  The displacement measures how far an object has moved from it’s starting position.  Note: ∆ is the Greek symbol “delta”. It means “a change in”.  x f means final position.  x i means initial position. Displacement: ∆x = x f – x i

The displacement of an object does not always equal the distance it has traveled.

Displacement can be either positive or negative…depending on your frame of reference. The right (or east) will be considered positive unless otherwise stated. Likewise with up (north). The left (or west) will be considered negative unless otherwise stated. Likewise with down (south). PositiveNegative ∆ x = 8 cm ∆ x = -4 cm

Displacment is only one portion of an object’s motion.  What if we also consider how long it took the object to change it’s position? Definition: average velocity – the total displacement divided by the total amount of time during the displacement. Velocity may be + or -, depending on the displacement. Velocity: v = ∆x = (x f -x i ) ∆t (t f – t i )

Why refer to it as average velocity? Example: Suppose you traveled from your house to school…a distance of 4.0 km. It took you.20 hours (12 minutes) because of heavy traffic. Your avg velocity would be: v = 4.0 km /.20 h v = 20 km/h Did you travel at that exact speed for the entire trip? Of course not. Because of this, velocity is an average.

Velocity and Speed are often used interchangeably.  Velocity and speed are not the same:  Velocity requires some direction. Speed refers to the numerical value (magnitude) of velocity.

Using the velocity equation, we can derive another equation to find an object’s position

Velocity can be determined using a position vs time graph. The slope of the line corresponds to the velocity.

Position vs Time Object 1 Object 2 Object 3 Time Position How could we describe the motion of these objects?

Chapter 2: Motion in One Dimension Section 2: Accelerated Motion

Definition: acceleration – the change in velocity over time.  An object accelerates when it changes its motion.  This means: » Speeding up » Slowing down » Changing direction Acceleration: a = ∆v/∆t = v f - v i t f - t i

Acceleration has the derived unit of m/s 2.  But…what does this unit actually mean? m 1 s x = _m_ s x s _m_ s 2 = m _s_ s =

 The magnitude of acceleration tells how quickly the change is happening.  The sign tells the direction.  Acceleration with a “+” magnitude means the object is gaining velocity in the “+” direction.  Acceleration with a “–” magnitude means the object is slowing down (or gaining velocity in the “–” direction.

 Objects that have a constant velocity have no acceleration.  If the velocity isn’t changing, then there is no acceleration. Definition: centripetal acceleration – the constant change of direction of an object moving in circles.  Although the magnitude of v may be constant, the object accelerates because it is changing direction.

Acceleration can be determined using a velocity vs time graph.

Velocity vs Time  The slope of a velocity vs time graph represents acceleration.

Using the base equation for acceleration, we can find other equations to find the velocity and/or displacement of moving objects that are changing their speed… With your partner: Using the base equation for acceleration, derive two new equations to solve for initial displacement and final velocity.

Create a position vs time AND a velocity vs time graph for the following scenario: A steel ball is placed at the top of a ramp, and allowed to roll down. The top of the ramp is the zero position. With your partner: Identify the relationship shown on each graph. Give the general equations for the relationships you identify.

Create a position vs time AND a velocity vs time graph for the following scenario: You roll a ball up a ramp with enough force to reach almost to the top. The ball then rolls back down the ramp. Work with your partner to identify these points on the velocity/acceleration graph: A point where the ball has a positive velocity. A point where the ball has zero velocity. A point where the ball has negative velocity.

Velocity with Constant Acceleration: Note that:  This equation does not require displacement.  ∆t = (t f – t i ) v f = v i + a∆t

Velocity with Constant Acceleration: Note that:  This equation does not require time.  ∆x = (x f – x i ) v f 2 = v i 2 + 2a∆x

Displacement with Constant Acceleration: Note that:  “t” is actually “∆t”.  However, t i is usually 0.  The equation can be rewritten to find x i. x f = x i + v i t + ½at 2

Displacement with Constant Acceleration: Note that:  This equation does not require acceleration. v f + v i 2 x =t

Falling Objects Definition: free fall – downward acceleration while under the effect of gravity only. Gravity is a force that causes objects to accelerate downward.

 All objects free fall with an acceleration of -9.8 m/s 2.  In other words, gravity causes objects to speed up as they fall downward.  In our studies, we will usually neglect air resistance until FRICTION is covered in more detail.

Practice Problems A race car traveling at 44 m/s accelerates to a velocity of 22 m/s over a period of 11 seconds. What was the car’s displacement during this time? 360 m.

Practice Problems A bike rider accelerates to a velocity of 7.5 m/s during 4.5 seconds. If the bike had a displacement of 19 m, what was its initial velocity?.9 m/s

Practice Problems A train accelerates from a velocity of 21 m/s with an acceleration of 3 m/s 2 over a distance of 535 meters. What is the final velocity of the train? 60 m/s