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SLIDE 1
Speed
Speed is the rate at which distance changes with time for a moving object.
? A train travels 300 km in 4 hours. What is the train’s average speed?
? A car travels 240 miles at an average speed of 60 miles per hour. How long does the journey take?
? How far does a car travel at an average speed of 65 kilometres per hour for 6 hours?
? Jack drives 115 miles at an average speed of 46 mph. If he starts his journey at 6 am, what time does he arrive at his destination?
SLIDE 2
? A train travels 300 km in 4 hours. What is the train’s average speed?
? A car travels 240 miles at an average speed of 60 miles per hour. How long does the journey take?
? How far does a car travel at an average speed of 65 kilometres per hour for 6 hours?
? Jack drives 115 miles at an average speed of 46 mph. If he starts his journey at 6 am, what time does he arrive at his destination?
SLIDE 3
? What is the average speed of a rocket that travels 3600 km in 45 minutes? Give your answer in
? How long does it take to travel 15 m at a speed of ?
? How far does a particle traveling at travel in 2 minutes? Give your answer in standard form.
SLIDE 4
? What is the average speed of a rocket that travels 3600 km in 45 minutes?
? How long does it take to travel 15 m at a speed of ?
? How far does a particle traveling at travel in 2 minutes?
SLIDE 5
Example
A motorist travelled a 37.5 mile section of motorway in 25 minutes. On the return journey he took 5 minutes longer to cover the same distance owing to a speed restriction. (a) What was the speed limit on the return? (b) What was the average speed to the nearest mile per hour of the entire journey?
Hint
Make a table of the information first.
The journey splits into two parts: 1 = the outbound journey; 2 = the return journey. Then there is T = the total journey. The speed is calculated in mph but the time here is given in minutes.
There are 60 minutes in an hour
25 minutes = ; 30 minutes = ,
Complete the following table, and so solve the problem.
Part of journey Speed (s) / mph Distance (d) / mi Time (t) / hr
1
2
T
SLIDE 6
A motorist travelled a 37.5 mile section of motorway in 25 minutes. On the return journey he took 5 minutes longer to cover the same distance owing to a speed restriction. (a) What was the speed limit on the return? (b) What was the average speed to the nearest mile per hour of the entire journey?
Solution
Part of journey Speed (s) / mph Distance (d) / mi Time (t) / hr
1 90
2 75
T 81.818… 75
SLIDE 7
A journey of 136 miles is made at an average speed of 32 mph. If the speed on the return journey is 40 mph, what is the average speed for the entire round trip?
SLIDE 8
A journey of 136 miles is made at an average speed of 32 mph. If the speed on the return journey is 40 mph, what is the average speed for the entire round trip?
Solution
Part of journey Speed (s) / mph Distance (d) / mi Time (t) / hr
1 32 136
2 40 136
T
272
The average speed is
SLIDE 9
A train travels outbound 264 miles at an average speed of 72 mph. On the return journey it takes 22 minutes less. What is the average speed of the return journey?
SLIDE 10
A train travels outbound 264 miles at an average speed of 72 mph. On the return journey it takes 22 minutes less. What is the average speed of the return journey?
Solution
Part of journey Speed (s) / mph Distance (d) / mi Time (t) / hr
1 72 264
2
264
The average speed of the return journey is 80 mph.
SLIDE 11
The distance from A to B is 45 miles. From B to C the distance is 20 miles. Georgia drives from A to B at an average speed of 30 mph and from B to C at an average speed of 40 mph. What is Georgia’s average speed for the whole journey?
SLIDE 12
The distance from A to B is 45 miles. From B to C the distance is 20 miles. Georgia drives from A to B at an average speed of 30 mph and from B to C at an average speed of 40 mph. What is Georgia’s average speed for the whole journey?
Solution
Part of journey Speed (s) / mph Distance (d) / mi Time (t) / hr
A to B 30 45
B to C 40 20
T
65
SLIDE 13
Peter and Paul cycled the same 50 km route. Peter took two-and-a-half hours to complete the journey. Paul started 5 minutes after Peter and overtook Peter when they had both cycled 15 km. They both cycled at constant speeds. What was Paul’s speed?
SLIDE 14
Peter and Paul cycled the same 50 km route. Peter took two-and-a-half hours to complete the journey. Paul started 5 minutes after Peter and overtook Peter when they had both cycled 15 km. They both cycled at constant speeds. What was Paul’s speed?
Solution
In this question, we find first the speed at which Peter is cycling, then the time it takes Peter to cycle the first 15 km. Then we find the time it took Paul to cycle the same 15 km, and finally the speed Paul was cycling.
Speed (s) /
Distance (d) / km Time (t) / hr
Peter
50
20 15
Paul
15 40 min
Paul cycled 22.5
SLIDE 15
A car travelled 200 miles at an average speed of 50 mph. On the return journey the speed increased by 10 percent. To the nearest minute, how much less time did the return journey take?
SLIDE 16
A car travelled 200 miles at an average speed of 50 mph. On the return journey the speed increased by 10 percent. To the nearest minute, how much less time did the return journey take?
Solution
Part of journey Speed (s) / mph Distance (d) / mi Time (t) / hr
1 50 200
2
200
The return journey took
3 hrs 38 min to the nearest minute
Less time
SLIDE 17
John’s position in race was recorded. His distance from the starting point was measured as metres, and his time from the start as t seconds.
When ; when . Plot these two points onto the graph. Find the gradient of this graph. What does the gradient show?
SLIDE 18
The gradient is
The gradient is the average speed of John during the race.
John’s average speed .
SLIDE 19
The Japanese are known to be fans of the writings of the Bronte sisters. A group of Japanese tourists visiting Yorkshire walked from their hotel to the Bronte Parsonage Museum.
The graph shows the distance of the tourist party from their hotel. Give your answers in where appropriate.
? What was the speed of the tourists as they walked towards the parsonage? (A)
? What happened during part B of the journey?
? What was their speed during part C?
? What happened between C and D? What was their speed during part D?
SLIDE 20
? A. They walked 4.5 km in 40 min, or of an hour. .
? B. They visited the parsonage.
? C. 2km in 30 mins.
? Between C and D they slowed down to 2.5 km in 50 min.
SLIDE 21
A group of generals were observing tests of three missiles.
Complete the table for the range, time and speed of each missile.
Missile Range / km Time / min Speed /
A
B
C
SLIDE 22
A group of generals were observing tests of three missiles.
Complete the table for the range and speed of each missile.
Missile Range / km Time / min Speed /
A 700 30
B 1000 64
C 1100 80
SLIDE 23
Distance and displacement
Suppose I walk in a straight line from my house 3 km to the post office and then turn around and walk straight back.
Fixing my house as the origin, by the time I have returned my position at my house, I am back at the origin.
However, the distance travelled is twice 3 km, or 6 km. I might be back home, but I’ve also walked 6 km.
Therefore, we have not one but two concepts (ideas).
A. The distance
B. The displacement from the origin.
Displacement has a direction – positive or negative from the origin. The choice of which direction is positive and which direction is negative is arbitrary. (This means, you or someone just decides.)
SLIDE 24
On Monday, Agent X flew 3,460 miles from London to New York. He returned 2 days later on Wednesday.
? Taking London as the origin, what was Agent X’s displacement before Agent X boarded the plane on Monday morning?
? Taking the positive direction as from London to New York, what was Agent X’s displacement on arrival at New York?
? Taking the positive direction as from New York to London, what was Agent X’s displacement on arrival at New York?
? What was the overall distance that Agent X travelled over the course of the whole journey?
? What was Agent X’s displacement on arrival at London on Wednesday?
SLIDE 25
On Monday, Agent X flew 3,460 miles from London to New York. He returned 2 days later on Wednesday.
? On Monday before boarding the plane, Agent X’s displacement was 0 (zero).
? Taking the positive direction as from London to New York, Agent X’s displacement on arrival at New York was .
? Taking the positive direction as from New York to London, Agent X’s displacement on arrival at New York was
? The overall distance that Agent X travelled over the course of the whole journey was . (This ignores any distance travelled while in New York.)
? Agent X’s displacement on arrival at London on Wednesday was again 0.
SLIDE 26
This graph shows the movement of an elevator (lift) in a vertical direction where 0 represents the ground floor.
? What do the positive and negative directions of displacement represent?
? Complete the following table.
Part Speed Direction
A
B
C
D
E
SLIDE 27
? Positive corresponds to floor levels above the ground floor; negative corresponds to basement levels.
?
Part Speed Direction
A
up
B 0
C
down
D 0
E
up
SLIDE 28
Velocity
In the preceding question we saw that the lift changes direction.
Velocity is speed and direction combined.
Part Speed /
Direction Velocity /
A
up
B 0
C
down
D 0 0
E
up
Speed has no sign . Velocity is a signed quantity.
SLIDE 29
The graph shows a displacement-time graph for the journey of a train between towns Zed, Yhy, Ex and Wyou. Complete the table.
Part Speed /
Direction Velocity /
A
B
C
D
SLIDE 30
Part Speed /
Direction Velocity /
A
B
C
D
SLIDE 31
? You are in a stationary train at a platform. The train starts, and 500s later you are travelling at . By this time, you have travelled a distance of 2 km.
What happened to your speed between the start and end of this motion? What do you feel in your back as you sit in the carriage facing the engine?
? Convert to giving your answer to 2 sf.
? Which of the following two graphs could be the displacement-time graph for this motion?
SLIDE 32
? You are in a stationary train at a platform. The train starts, and 500s later you are travelling at . By this time, you have travelled a distance of 2 km.
Your speed began at and you gradually speeded up to reach a speed of . During the first part of the journey you feel the back of the seat “pushing” you. The force exerted by the engine is causing you to speed up.
?
? Graph A could be the displacement-time graph for this motion.
Graph A shows that you were speeding up for the first part of the journey. The second graph suggest that you started immediately at a constant speed. The constant speed in graph B is which is not the final speed of .
SLIDE 33
Convert
? to ? to
? to ? to
SLIDE 34
?
?
?
?
SLIDE 35
Which of the following graphs represents a constant speed
of ? (More than one answer is possible.)
SLIDE 36
In all these graphs, the gradient is the same and constant at . All four of them represent a constant speed of .
SLIDE 37
Complete the table
Change in displacement Change in time Velocity
/
A
B
C
D
SLIDE 38
Change in displacement Change in time Velocity
/
A 100 10 10
B 100 20 5
C
12 5
D
10
SLIDE 39
Velocity and the gradient of the displacement-time graph
An upward sloping graph indicates a positive velocity. A downward sloping graph indicates a negative velocity.
Find the velocity of the object in each case.
SLIDE 40
A The gradient is upward sloping. The velocity is positive.
B The gradient is downward sloping. The velocity is negative.
SLIDE 41
Acceleration is change in velocity. Acceleration is speeding up in a direction, and deceleration is slowing down in a direction.
Match each of these graphs to a description.
? Constant velocity of
? Negative acceleration to a velocity of
? Constant speed of followed by deceleration (negative acceleration) to .
? Acceleration from rest to a constant speed of .
SLIDE 42
A ? ? Constant speed of followed by deceleration (negative acceleration) to
B ? ? Acceleration from rest to a constant speed of
C ? ? Constant velocity of
D ? ? Negative acceleration to a velocity of
SLIDE 43
Choice of direction
Displacement, velocity and acceleration are all signed quantities.
In one-dimension they may be positive or negative .
The choice of which direction shall be positive is arbitrary – meaning that someone chooses. Often, we think of the positive direction as running from left to right, but we can swap the directions around. Also, the positive direction could be up or down.
Imagine you are in a car and the car speeds up as it goes onto a motorway. Then, the car slows down as it leaves the motorway. The speeding up is positive acceleration, and the slowing down, which is acceleration in the opposite direction is negative acceleration.
Because it is slowing down, we also call this negative acceleration, deceleration or retardation. This can be confusing, though, as negative acceleration will eventually cause an object to reverse the direction of its motion and accelerate in the negative direction. So, we use the words deceleration and retardation in cases when the velocity of the object can only be positive, so that when an object slows down it never changes direction. Its acceleration may be negative, but not its velocity. To decelerate means to slow down while moving in the positive direction.
SLIDE 44
Throw a pebble straight up and catch it. What do you observe?
In this question ignore the effect of air resistance.
? Why doesn’t the pebble carry on travelling upwards forever?
? Why does the pebble change direction?
? If up is taken to be the positive direction, what is the direction of the pebble after it has reached its maximum height?
? Before the pebble has reached its maximum height, when is it travelling the fastest?
? In which direction is the pebble accelerating (a) before it reaches its maximum height, (b) after it reaches its maximum height?
? After the pebble has reached its maximum height, when is it travelling the fastest?
SLIDE 45
Throw a pebble straight up and catch it. What do you observe?
In these answers we ignore the effect of air resistance.
? Gravity causes the pebble to move towards (accelerate towards) the Earth.
? The pebble changes direction because gravity causes it to move towards the Earth. The pebble is not fitted with something like an engine to enable it to thrust itself constantly upwards.
? Taking up to be the positive direction of the movement, after the pebble reaches its maximum height, it is travelling in the negative direction.
? Before the pebble has reached its maximum height, it is travelling fastest when it leaves your hand. You give the pebble a thrust or impulse that causes it to leave your hand with maximum velocity. After that, the pebble first slows down until it reaches its maximum height, and then changes direction and accelerates back towards the Earth.
? The pebble both (a) before it reaches its maximum height, (b) after it reaches its maximum height is always accelerating towards the Earth in the negative direction.
? After the pebble has reached its maximum height, the pebble is travelling fastest when you catch it in your hand. Once the pebble has reversed direction (a moment of instantaneous rest), it speeds up until you catch it.
SLIDE 46
A pebble is thrown straight up and returns exactly to the position from whence it was launched. Ignore the effect of air resistance.
Which of the above graphs could represent the motion of the pebble?
SLIDE 47
A pebble is thrown straight up and returns exactly to the position from whence it was launched. Ignore the effect of air resistance.
The motion of the pebble could be represented by D. Here the pebble starts and ends with maximum velocity, so has the steepest gradient. It slows down to a moment of instantaneous rest and changes direction.
SLIDE 48
Newton’s first law
A smooth surface is a surface where there is no friction at all.
An object that is moving on a smooth surface will continue moving with the same velocity forever, unless it collides with something.
(This is a part statement of Newton’s first law.)
SLIDE 49
A smooth surface is enclosed by its sides. An object travelling in the positive direction collides with one of the sides and instantaneously changes its motion to the negative direction with its speed not altered.
Which of the above graphs could represent the motion of the object?
SLIDE 50
A smooth surface is enclosed by its sides. An object travelling in the positive direction collides with one of the sides and instantaneously changes its motion to the negative direction.
Graph A could represent this object. It travels with constant velocity in the positive direction shown by the straight line with constant gradient, and then instantaneously changes direction to move with the same speed in the negative direction. The size of the gradient is unchanged, but the gradient is now sloping downwards.
SLIDE 51
A rubber ball is elastic.
Observe what happens when a rubber ball is dropped vertically downwards. Does the ball bounce back to the same height from whence it was dropped?
Observe what happens when a lump of putty is dropped vertically downwards. Does the lump of putty bounce?
Explain in terms of energy these observations.
SLIDE 52
When a rubber ball is dropped vertically downwards it speeds up until it reaches the floor. At the floor it bounces and moves in the reverse direction. It does not bounce back to the same height from whence it was dropped.
A lump of putty simply stops when it reaches the floor.
The rubber ball gains energy as it accelerates towards the floor, speeding up. When it bounces, some of this energy is lost. (This lost energy is absorbed by the floor and converted to heat.) Because the ball is elastic, it bounces, but because it has lost some of its energy, it does not return to the same height as before.
As it falls the lump of putty also speeds up, but when it makes contact with the floor all of this energy is lost.
SLIDE 53
A collision is perfectly elastic if no energy is lost during the collision.
If a rubber ball is perfectly elastic, when it is dropped from a height, it will bounce back to the same height.
Sketch the graph of the height (displacement) of a rubber ball when dropped (a) for a perfectly elastic collision with the floor, and (b) for an elastic but not perfectly elastic collision with the floor.
Include three bounces.
SLIDE 54
Three bounces of a perfectly elastic rubber ball.
Three bounces of a not perfectly elastic rubber ball.
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