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SCALARS? A scalar is a quantity which has magnitude (number value) only Examples of scalars are speed, distance, energy, charge, volume, mass, weight and temperature. The value of a scalar is called its magnitude. Scalar quantities can be directly added or subtracted from each other

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VECTORS? A vector is a quantity which has both a magnitude and a direction. Examples of vectors are displacement, velocity, and acceleration. Vectors cannot be added in the same way as scalars!

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VECTOR NOTATION Vectors are represented by an arrow on top of the symbol. Below is an example of velocity, which is a vector. v

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Hold up either “S” or “V” Tony runs 80 m

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Hold up either “S” or “V” Sheri throws the ball 20m up

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When a direction is written in a vector description, it is usually abbreviated and put into square brackets. [W] is west[E] is east [N] is north[S] is south

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DIRECTION & DISTANCE Distance: Length of path which a body covers during motion Units: metre (m) or kilometer (km) Displacement: The change in position of an object during motion. Units: metre (m) or kilometer (km) Distance is a scalar, and displacement is a vector variable.

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Displacement (blue line) is how far the object is from its starting point, regardless of how it got there. Distance traveled (dashed line) is measured along the actual path.

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When calculating displacement North is positive, South is negative East is positive, West is negative Up is positive, Down is negative Right is positive, Left is negative

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DISPLACEMENT FORMULA The formula for displacement is: This formula means the “change in position” is equal to the final position minus the initial position. The answer is written with the magnitude first, and the direction in square brackets second. For example: 40 km [E] ddd i f - =

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DISPLACEMENT FORMULA EXAMPLE John rode his bicycle from point that was 3 km west of a point and stopped at a place located 12 km east of that point. Calculate his displacement. Answer: 15 km [E] 3km12km

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MATHEMATICAL EXAMPLE OF DISTANCE AND DISPLACEMENT #2 Wendy rides her bike 10km North. Then she goes 8 km East and finally 4 km South. What is Wendy’s Displacement? What is Wendy’s Distance? 10km [NE]6km [N], 8km [E] 22km

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A NOTE WITH DISPLACEMENT WATCH FOR SIGNS (+ OR - ) CHOOSE A REFERENCE DIRECTION AS POSITIVE BE CONSISTENT WITH YOUR SIGNS!

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TIME INTERVALS Initial Time is the time when an event starts. Final Time is the time when the event ends. The time interval ∆t is the total time of the event. Note: ∆ means change t t ttt f i i f - =

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CONCEPTUAL TIME EXAMPLE #1 A show starts at 8:25 and ends at 8:52. What is the shows time interval? 27 min

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WHAT IS UNIFORM MOTION Uniform motion is when you travel equal displacements in equal time intervals This rarely happens in real life, but it is a good approximation.

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Which one is Uniform Motion? X X X X X X X X X X X X X X X

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Describes the rate of change in position of an object over time

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The slope is equivalent to the average speed

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A constant slope indicates uniform motion.

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The reference point (origin) is needed so we know which way is positive, which way is negative, and where zero is. Reference point is where the x and y axis intersect Reference point

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Positive Slope: The object is moving in the positive direction, and its position is increasing with time. This also means that the object has a positive velocity (has a speed in the positive direction)

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Negative Slope: The object is moving in the negative direction, and its position is decreasing with time. This also means that the object has a negative velocity (has a speed in the negative direction)

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Zero Slope: The object is at rest. The object is not moving. The position is not changing.

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If an object is staying in the same position for a time interval than its… at rest stopped not moving still Represented by a horizontal line on a position vs time graph

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2) When and where is the object not moving? position time 1 m 2 m 2 s 6 s4 s 8 s 10 s 3 m 4 m 5 m 0 s 0 m

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1)The object is at rest at 4 m position time 1 m 2 m 0.5 s 1.5 s1.0 s 2.0 s 2.5s 3 m 4 m 5 m 0 s 0 m

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The object who turned around at 4 seconds position time 1 m 2 m 2 s 6 s4 s 8 s 10 s 3 m 4 m 5 m 0 s 0 m

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SPEED AND VELOCITY, WHATS THE DIFFERENCE? Can two objects have the same speed and different velocities? Velocity is a vector – has direction and magnitude Speed is a scalar, it only has magnitude. (move in different directions) YES

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AVERAGE VELOCITY Average velocity is equal to the change in position (displacement) divided by the change in time. This is why velocity has the units (m/s)

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AVERAGE VELOCITY In a way, average velocity is a simplification. It tells us on average over a certain time interval how fast an object is moving. It does not tell us exactly what velocity the object moved at for all instants in that interval.

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DIFFERENCE BETWEEN AVERAGE VELOCITY AND AVERAGE SPEED. I walk 10 m North in 5 seconds and then walk for 10 m South in the next 5 seconds. What is my average velocity? My average speed? 0 m/s 20m = 2 m/s 10s

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Calculating Average Velocity The relationship between average velocity, displacement, and time is: 1. What is the average velocity of a dog that takes 4.0 s to run forward 14 m? (3.5 m/s forward) 2. A boat travels 280 m east in a time of 120 s. What is the boat’s average velocity? (2.3 m/s east)

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CONVERTING UNITS: m/s km/h For example, convert 75 km/h to m/s.

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PROBLEM #1 What is the average velocity of a skateboard that goes 50 m [W] over a time interval of 8 seconds? Answer: Average Velocity = 6.25 m/s [W]

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PROBLEM #2 A deer is running forward at a speed of 46 km/h. How far does it travel in 30 seconds? Displacement = km or 384 meters

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PROBLEM #3 How many minutes would it take a car moving at an average velocity of 82 km / h to travel 125 km? Time = 91.5 minutes

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Acceleration Acceleration (a) is the rate of change in velocity (how fast velocity changes) This change in velocity can be due to a change in speed and/or a change in direction. Two objects with the same change in velocity can have different accelerations. This is because acceleration describes the rate at which the change in velocity occurs. Suppose both of these vehicles, starting from rest, speed up to 60 km/h. They will have the same change in velocity, but, since the dragster can get to 60 km/h faster than the old car, the dragster will have a greater acceleration.

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Positive and Negative Acceleration The direction of the acceleration is the same as the direction of the change in velocity. Acceleration that is opposite the direction of motion is sometimes called deceleration.

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Acceleration can be described in a velocity-time graph. BE CAREFUL not to confuse this with a position-time graph! Examples of acceleration: 1. A car speeding up in the forward direction If we designate the forward direction as positive (+), then the change in velocity is positive (+), therefore the acceleration is positiv e (+).

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Examples of acceleration: 2. A car slowing down in the forward direction. If we designate the forward direction as positive (+), then the change in velocity is negative ( - ), therefore the acceleration is negative ( - ).

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Examples of acceleration: 3. A car speeding up in the backward direction. If we designate the backward direction as negative ( - ) then the change in velocity is negative ( - ). This means that the acceleration is negative ( - ) even though the car is increasing its speed. Remember positive (+) and negative ( - ) refer to directions!!!!

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Examples of acceleration: 4. A car slowing down in the backward direction. If we designate the backward direction as negative ( - ) then the change in velocity is positive (+). This means that the acceleration is positive (+) even though the car is decreasing its speed. Remember positive (+) and negative ( - ) refer to directions.

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Zero Acceleration Examples of acceleration: 5. A car is stopped. The change in velocity is zero. The car is at rest. This means that the acceleration is zero.

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Examples of acceleration: 6. A car has constant velocity (uniform motion). The change in velocity is zero. This means that the acceleration is zero. The car is not speeding up or slowing down, but moving at constant velocity.

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Calculate (1) Bus is moving 15 m/s backwards and slows down to 5 m/s backwards. (2) Train is moving 300 km/h forwards and speeds up to 400 km/h forward. (3) Car starts at 5 m/s forwards and finishes at 10 m/s backwards. (4) Jeep is moving 8 m/s backwards and slows to a stop.

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9.2 Calculating Acceleration The acceleration of an object depends on the change in velocity and the time required to change the velocity. When stopping a moving object, the relationship between time and acceleration is: Increasing the stopping time decreases the acceleration. Decreasing the stopping time increases the acceleration.

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Velocity-Time Graphs The motion of an object with a changing velocity can be represented by a velocity-time graph. The slope of a velocity-time graph is average acceleration. Acceleration is measured in m/s 2.

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Determining Motion from a Velocity-Time Graph A velocity-time graph can be analyzed to describe the motion of an object. Positive slope (positive acceleration) – object’s velocity is increasing in the positive direction Zero slope (zero acceleration) – object’s velocity is constant Negative slope (negative acceleration) – object’s velocity is decreasing in the positive direction or the object’s velocity is increasing in the negative direction

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Calculating Acceleration The relationship of acceleration, change in velocity, and time interval is given by the equation: Example: A pool ball travelling at 2.5 m/s towards the cushion bounces off at 1.5 m/s. If the ball was in contact with the cushion for 0.20 s, what is the ball’s acceleration? Assume towards the cushion is the positive direction. (+)

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Calculating Acceleration Try the following acceleration problems. 1. A truck starting from rest accelerates uniformly to 18 m/s [W] in 4.5 s. What is the trucks acceleration? 2. A toboggan moving 5.0 m/s forward decelerates backward at m/s 2 for 10 s. What is the toboggans velocity at the end of the 10 s? 3. How much time does it take a car travelling south at 12 m/s to increase its velocity to 26 m/s south if it accelerates at 3.5 m/s 2 south?

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9.2 Gravity and Acceleration

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Gravity and Acceleration Objects near the surface of Earth fall to Earth due to the force of gravity. Gravity is a pulling force that acts between two or more masses. Air resistance is a friction-like force that opposes the motion of objects that move through the air. Ignoring air resistance, all objects will accelerate towards Earth at the same rate. The acceleration due to gravity is 9.8 m/s 2 downward.

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Calculating Motion Due to Gravity To analyze situation where objects are accelerating due to gravity, use the equations: In these equations, the acceleration ( ) is 9.8 m/s 2 downward. Example: Suppose a rock falls from the top of a cliff. What is the change in velocity of the rock after it has fallen for 1.5 s? Assign “down” as negative (-).

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Calculating Motion Due to Gravity Try the following acceleration due to gravity problems. 1. What is the change in velocity of a brick that falls for 3.5 s? 2. A ball is thrown straight up into the air at 14 m/s. How long does it take for the ball to slow down to an upward velocity of 6.0 m/s? 3. A rock is thrown downwards with an initial velocity of 8.0 m/s. What is the velocity of the rock after 1.5 s?

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