Motion part 1 1 Scalar quantities versus

Motion (part 1) 1

Scalar quantities versus vector quantities Scalar quantities have a size (or magnitude) only. Eg: Speed of the car was 60 km/h Vector quantities have both size (magnitude) and direction. Eg: The boat sailed 30 km north. 2

How do we draw a vector? As an arrow !! the length of the arrow represents the size of the vector The direction the arrow points indicates the direction of the vector! When we draw vectors we need to include a reference direction and a scale (e. g. 1 cm = 1 km) N 4. 3 km North west 3

Distance and Displacement 4

Distance is a scalar quantity! is how far something has moved. is measured in meters etc. Displacement is a vector quantity. is how far an object has moved from its starting point and in what direction. measured in meters but also has a direction. is the (straight line) distance between 2 points with the direction given too! 5

If the bear skis the 7 meters to the tree then back to the start and then all the way to the house, what distance would he have gone? _______m 6

What would his displacement from the starting point be ? ______ m in an easterly direction. 7

Speed and velocity 8

Speed is a scalar quantity! It is how fast something has travelled It is measured in meters/second or km/hour. Velocity is a vector quantity. This is how fast and in what direction something travels IN A STRAIGHT LINE!!! This is measured in m/s or km/h but also has a direction. 9

Speed If a car travels 150 km in 2 hours then calculate the average speed of the car: Average Speed = distance travelled/ time = d/t = 150 km/ 2 h = 75 km/h Or write it like this: = 75 km h ⁻¹ 10

Velocity If a plane travels 480 km in a southerly direction for 2 hours then calculate the average velocity of the plane: Average velocity= distance travelled in a straight line time Vav = s/t = 480 km/ 2 h South = 240 km/h South Or write it like this: = 240 km h ⁻¹ South 11

How do we convert a speed or velocity from km/h to m/s or the other way around? How many meters in a km? 1000 m. How many seconds in an hour? Mmmmm, a bit harder… 60 seconds in a minute, 60 minutes in an hour so 60 x 60 = 3600 seconds in an hour. 1 km/hr = 1000/3600 m/s 75 km/h = 75 x 1000/3600 m/s = 20. 8 m/s 10 m/s = 10 x 3600/1000 km/h = 36 km/h 12

A different way of writing these: You are used to seeing “kilometers per hour” written as km/h but it is written ‘scientifically’ in the form: km h-1 And meters per second can be written as m/s or better still: m s-1 13

A canoe travels a distance of 500 m in 100 sec. What is the canoe's speed? speed = d/t A second canoe travels a distance of 200 meters in 40 seconds. What is this canoe's speed? 14

Remember that if an object is going at a constant speed but changes direction (turns) then its velocity changes!! 15

All these abbreviations – what do they stand for? ? ? !!! d = distance s = displacement t = time v= velocity or final velocity u = initial velocity a = acceleration 16

Acceleration • Acceleration is a vector quantity. • When an object increases velocity (speeds up) we say that it is accelerating. • When an object is slowing down we say it has negative acceleration, deceleration or retardation. 17

How do we calculate acceleration (negative or positive)? Acceleration = final velocity – initial velocity time Units = m/s 2 or m s-2 or meters per second. a = Vf – Vi t a = acceleration, Vf = final velocity, Vi = initial velocity, t = time. 18

Now lets try an example. A B At point A , the bike is going at 10 ms-1 , 5 seconds later at point B , the bike is going at 45 ms-1 What acceleration does the bike have between A and B? a = (v – u)/ t = (45 – 10) / 5 = 35/5 = 7 ms-2. Which means it is speeding up: each second it goes 7 m/s faster than the second before! 19

Now lets try another example. C D At point C , the bike is going at 45 ms-1 , 10 seconds later at point D , the bike is going at 20 ms-1 What acceleration does the bike have between C and D? a = (v – u)/ t = (20 – 45) / 10 = -25/10 = - 2. 5 ms-2. Which means it is slowing down: each second it goes 2. 5 m/s slower than the second before! 20

A B At A the camel has a velocity of 3 m/s, it accelerates at 2 m/s². What is its velocity 4 seconds later at B? a = (v – u) t So rearrange: v = u + at = 3 +(2 x 4) =3+8 = 11 m/s 21

If we took photos of the camel at 1 second intervals before he started accelerating, the series of pictures would look like this: t=1 s t=2 s t=3 s t=4 s t=5 s Notice that the intervals between the positions are equal. This is because the camel was going at a constant velocity (no acceleration!!!) 22

We now take photos of a runner at 1 second intervals with an initial velocity of 2 m/s and acceleration of 1 m/s/s. (She starts accelerating at t = 1 seconds. ) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 distance in meters Notice that the intervals between the positions are increasing by 1 m each second. This is because the runner was accelerating at 1 m/s/s. (Is this a realistic acceleration? ) 23

Another equation!!! Sometimes we need to work out how far an object has moved – or been displaced - when all we know is its initial velocity, final velocity and acceleration!!! To work this out we use this equation: s = Vit + ½ at 2 Where s = displacement, t = time, Vi= initial velocity and a = acceleration. 24

Notice that we can rearrange this equation and use it to work out other things such as initial velocity! Lets look at an example: A thief jumps in out of a window and lands on a taxi going past. The taxi driver gets a fright and accelerates at 7 m/s/s. If the taxi takes 4 seconds to travel 90 meters before crashing into a vegetable cart, what was its initial velocity when the thief landed on it? 25

What do we know? s = 90 m, t = 4 s, a = 7 m/s 2, u = ? Rearrange: s = ut + ½ at 2 u = (s – ½ at 2) / t u = (90 – ½ x 7 x 42) / 4 u = 8. 5 m/s 26

Motion graphs It is often easier to show the motion of an object with a graph rather than with words. There are 2 types of graph we will look at: Displacement– time graphs (or distance – time graphs. ) Velocity - time graphs (or speed- time graphs. ) 27

• With both types of graph, time is plotted on the x axis. • The further to the right along the x axis we go – the longer the time from the start! • Velocity, distance etc are always plotted on the y axis. • We assume the initial direction of motion to be positive. 28

0 1 Distance (m) 2 3 4 5 6 7 8 9 10 11 A body at rest. Ie it is stopped or standing still! 0 1 2 3 4 5 6 7 time (s) 8 9 10 11 12 29

0 1 Distance (m) 2 3 4 5 6 7 8 9 10 11 An object moving at constant speed. Constant speed because the lines are straight! Which line shows the object going fastest? 0 1 2 3 4 5 6 7 time (s) 8 9 10 11 12 30

0 1 Distance (m) 2 3 4 5 6 7 8 9 10 11 Speed is given in m/s so we can work out the speed here by saying speed = rise/run = distance/time. Work it out for each. 0 1 2 3 4 5 6 7 time (s) 8 9 10 11 12 31

0 1 Distance (m) 2 3 4 5 6 7 8 9 10 11 A body accelerating! You can see that the speed is increasing: the distance travelled is more each second so this shows it is accelerating!! 0 1 2 3 4 5 6 7 time (s) 8 9 10 11 12 32

Lets look at an example: First we will give the information in words then as a displacement – time graph. Mark starts from point A and travels at 2 m/s for 3 seconds to point B. He then stops at point B for 4 seconds before going back towards point A at an initial velocity of -1. 5 m/s for 2 seconds then stopping at an intersection for 1 second before continuing to point A at -1. 5 m/s. Ok – lets look at this graphically!!!!! 33

Much Easier than words!!! 0 1 Distance (m) 2 3 4 5 6 7 8 9 10 11 Marks’ trip 0 1 2 3 4 5 6 7 time (s) 8 9 10 11 12 34

8 9 10 11 What is the difference between this graph and the ones we looked at before? ? 0 1 Speed (m/s) 2 3 4 5 6 7 This axis has SPEED not distance on it!!!! 0 1 2 3 4 5 6 7 time (s) 8 9 10 11 12 35

9 10 11 So what does this graph show? 0 1 Speed (m/s) 2 3 4 5 6 7 8 It shows that the object is moving at a constant speed of 5. 5 m/s 0 1 2 3 4 5 6 7 time (s) 8 9 10 11 12 36

9 10 11 What does this graph show? 0 1 Speed (m/s) 2 3 4 5 6 7 8 It shows that the objects’ speed is increasing or the object is accelerating! 0 1 2 3 4 5 6 7 time (s) 8 9 10 11 12 37

9 10 11 What does this graph show? 0 1 Speed (m/s) 2 3 4 5 6 7 8 It shows that the objects’ speed is decreasing or the object is decelerating! 0 1 2 3 4 5 6 7 time (s) 8 9 10 11 12 38

Which line shows faster acceleration? Green or blue? 0 1 Speed (m/s) 2 3 4 5 6 7 8 9 10 11 Acceleration = speed/time so Blue = 3 m/s/s Green = 1 m/s/s Steeper slope = faster acceleration! 0 1 2 3 4 5 6 7 time (s) 8 9 10 11 12 39

9 10 11 Putting it all together: 8 Steady speed Fast deceleration Or negative acceleration Speed (m/s) 2 3 4 5 6 7 Fast accn 0 1 Slow accn 0 1 2 3 4 5 6 7 time (s) 8 9 10 11 12 40

Distance (m) 10 20 30 40 50 60 Speed (m/s) 1 2 3 4 5 6 7 8 What other information can we get from displacement – time and speed time graphs? 1 2 3 4 5 6 7 8 9 Time (s) Gradient = rise/run = 30/6 = 5 m/s = speed! 1 2 3 4 5 6 7 8 9 Time (s) Gradient = rise/run = 7/7 = 1 m/s/s = acceleration!!! Area under the graph = ½ time x speed = s x m/s =m = distance travelled!!! 41

Lets look at the last example and work out the distance travelled. 42

0 1 Speed (m/s) 2 3 4 5 6 7 8 9 10 11 Work out the area under the line to work out the distance travelled 0 1 2 3 4 5 6 7 time (s) 8 9 10 11 12 43

The end…. . Test Time!! 44