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Acceleration and the velocity time graph
SLIDE 1
Consolidation
? What is the difference between distance travelled and displacement?
? Define speed and velocity.
SLIDE 2
? What is the difference between distance travelled and displacement?
Solution
Distance is how far an object travelled in any direction.
Displacement is how far an object is from a point of reference, called the origin.
? Define speed and velocity.
Solution
SLIDE 3
The graph shows the displacement of a train relative to its starting point up and down a straight track.
? For how long altogether was the train at rest?
? What was the speed and velocity of the train during part A of the journey?
? What was the speed and velocity of the train during part C of the journey?
SLIDE 4
? The train was at rest during parts B and D of the journey for a total of 30 min.
? Part A, .
The velocity is .
? Part C, .
The velocity is .
SLIDE 5
Scalars and vectors
A scalar is any quantity that has size or magnitude only.
A vector is any quantity that has both size and direction.
Distance and speed are scalars.
Displacement and velocity are vectors.
SLIDE 6
In each of the following one is a scalar and the other is a vector. Determine which is which.
? Volume of liquid in a container. Rate at which the volume is changing because of liquid flowing in or out.
? Rotation of an angle. The size of an angle.
? Weight and mass.
? The density of a block of wood. The pressure exerted by the block of wood on the surface of a table.
SLIDE 7
? Volume of liquid in a container ? scalar
Rate at which the volume is changing because of liquid flowing in or out ? vector, can flow in or out .
? Rotation of an angle ? vector, can be clockwise or anticlockwise*.
The size of an angle ? scalar, as indicated by the word “size”.
? Weight ? vector. Weight is the force of gravity acting on a mass, and acts “downwards” towards, for example, the centre of the Earth.
Mass ? scalar. It is a measure of the quantity of substance or matter, and cannot be negative.
? The density of a block of wood ? scalar. Density is the ratio of mass to volume (“mass per unit volume”). It is a ratio of two scalars, so is a scalar.
The pressure exerted by the block of wood on the surface of a table ? vector. Pressure is force per unit area. Area is a scalar, but force is a vector. Pressure has a direction.
SLIDE 8
Sketch the displacement time (km against min) graph for a train that travels 40 minutes at , is then stationary for 10 minutes, and then returns to its original starting position at a speed of .
SLIDE 9
For the first part of the journey, 40 min at
For the last part of the journey
SLIDE 10
In which of the following graphs is there a constant velocity of ? (More than one answer is possible.)
SLIDE 11
For a negative velocity, the graph must be downward sloping, and only in A is the gradient .
SLIDE 12
What is the relationship between velocity and the displacement-time graph?
SLIDE 13
Velocity is the gradient of the displacement-time graph.
SLIDE 14
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 1000 m.
? What happened to your velocity between the start and end of this motion?
? What do you feel in your back as you sit in the carriage facing the engine?
? What is acceleration?
SLIDE 15
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 1000 m.
? Between the start and end of this motion your velocity has changed from to .
? As you sit in the carriage facing the engine you feel a force acting on your back, pushing you forward. Forces cause objects to change their velocity.
? Acceleration is a change in velocity.
SLIDE 16
The instantaneous velocity is the gradient of the tangent to the displacement-time graph.
The method is approximate, since the position of the tangent is uncertain.
At 35s the instantaneous velocity of this object is .
SLIDE 17
A stationary train starts at . The following displacement-time graph represents the motion of this train.
? Find the instantaneous velocity of the train at 20s and 40s.
? What is the change of velocity of the train between 20s and 40s?
SLIDE 18
? At 20s the train is traveling with velocity .
At 40s the train is traveling with velocity
? The change in velocity is approximately
SLIDE 19
Average acceleration between two times is
Example
At 20s a train was traveling with velocity . At 40s the same train was traveling with velocity . What was the average acceleration of the train?
Solution
The units of acceleration are metres per second per second, written . Sometimes, is still used.
SLIDE 20
At time a train was traveling with a velocity of . At time it was traveling with a velocity of . What was its average acceleration?
SLIDE 21
At time a train was traveling with a velocity of . At time it was traveling with a velocity of . What was its average acceleration?
Solution
SLIDE 22
Notation for changes
This is written in symbols
Here the Greek symbol, (capital delta, Greek for “d”), is used to stand for a change. The symbol is read “change in velocity”.
SLIDE 23
? It is given that is the equation of a straight line.
Translate into words.
? Write in words, where E stands for energy, m for mass, c is a constant and (Greek, theta) stands for temperature.
SLIDE 24
? It is given that is the equation of a straight line.
Translate into words.
Solution
? Write in words, where E stands for energy, m for mass, c is a constant and (Greek, theta) stands for temperature.
Solution
SLIDE 25
Use of subscripts
We have occasionally used subscripts before.
Example
The velocity of an object at time is
Its velocity at time is
Its average acceleration is
? Multiply top and bottom of by . What do you observe about the order of the subscripts in this expression?
? A straight line passes through the points and . Using these points find an expression for the gradient of the line, where
SLIDE 26
?
The order in which you input the values does not change the acceleration. This is because the signs cancel out.
? A straight line passes through the points and . The equation for the gradient is
SLIDE 27
In each case, calculate the average acceleration.
If the answer is not exact, give it to 3 significant figures.
?
?
?
?
SLIDE 28
?
?
?
?
SLIDE 29
The velocity-time graph
The figure shows the velocity-time graph for a car.
? What is the velocity at ?
? What is the velocity at ?
? What is the constant acceleration of the car?
SLIDE 30
? At ,
? At ,
?
SLIDE 31
Constant acceleration is also known as uniform acceleration
The diagram shows the velocity-time graph for a cyclist.
? During which time periods did the cyclist have positive acceleration?
? During which time periods was the cyclist decelerating?
? When was the cyclist at rest?
? When was the cyclist travelling with uniform acceleration?
? When, if at all, was the speed of the cyclist constant?
? During which time periods was the cyclist’s acceleration non-uniform?
SLIDE 32
? During which time periods did the cyclist have positive acceleration? ? Periods A, B and C.
? During which time periods was the cyclist decelerating?
? Periods E and F.
? When was the cyclist at rest? ? At and
? When was the cyclist travelling with uniform acceleration?
? Periods B, D and F. (During D, the acceleration is zero.)
? When, if at all, was the speed of the cyclist constant? \cv
? Period D.
? During which time periods was the cyclist’s acceleration non-uniform? ? Periods A, C and E. (During E, non-uniform deceleration.)
SLIDE 33
The graph shows the speed of Charlie running on a treadmill. Which of the following statements is false?
A Charlie ran for a total of 34 minutes.
B Charlie accelerated for total of 14 minutes.
C Charlie spent more time decelerating than he did accelerating.
D Charlie ran at a constant speed for 6 minutes.
SLIDE 34
A Charlie ran for a total of 34 minutes.
B Charlie accelerated for total of 14 minutes.
C Charlie spent more time decelerating than he did accelerating ? FALSE
Charlie decelerated for 14 minutes – the same time that he spent accelerating.
D Charlie ran at a constant speed for 6 minutes.
SLIDE 35
The diagram is a velocity-time graph for the flight of a bird.
? For how long was the bird flying, and what was the bird’s velocity during this period?
? What distance in km did the bird fly?
SLIDE 36
? The bird flew for a total of at a constant velocity of .
?
SLIDE 37
Distance travelled is the area under the velocity-time graph
SLIDE 38
The diagram shows the flight of a drone.
? What was the acceleration of the drone in the first part of the journey?
? What was the deceleration of the drone in the last part of the journey?
? Find the distance travelled by the drone throughout the journey.
SLIDE 39
? The acceleration of the drone in the first part of the journey
? The deceleration of the drone in the last part of the journey
? The distance travelled by the drone throughout the journey.
This is the area under the graph. This area may be seen either as comprising a rectangle and two triangles, or as a trapezium.
SLIDE 40
The diagram shows the journey of a train on a track. What was the total displacement and total distance travelled of the train by the time it reached its destination 140 mins after the start of journey?
SLIDE 41
In this question the positive displacement cancels out the negative displacement. The train travels a certain distance in one direction, then turns around and travels a distance in the reverse direction.
SLIDE 42
Find the total displacement and the total velocity.
SLIDE 43
SLIDE 44
Estimation of the area under a curve
The area under a curve can be estimated (approximated) by finding the area of rectangles.
The above diagram is the velocity-time graph of a rocket. An estimation for the distance travelled by the rocket in the first 60 s has been made by fitting rectangles to the mid-points. Two estimates of the velocities have been added. Complete the diagram and estimate the distance travelled.
SLIDE 45
SLIDE 46
By comparing the size of the two shaded parts in the diagram, explain why the above method of estimating an area may be regarded as “quite good”.
SLIDE 47
In the above diagram the area of the pink shaded part is included from the estimate of the area under the curve, and the area of the blue shaded included. By eye the area that is included is about the same as the area that has been excluded, so the two errors in the estimation approximately cancel out. Thus, the overall estimate of the area is “quite good”, which is a subjective judgement.
SLIDE 48
Velocity-time graph of a rocket
We sometimes deliberately find an underestimate of the area under a curve. This underestimate comprises rectangles that just touch the curve at the corner. Complete the estimate for the distance travelled by the rocket in the first 60 s.
SLIDE 49
Velocity-time graph of a rocket
The underestimate of the distance travelled by the rocket is
SLIDE 50
Greatest lower bound
The underestimate of the distance travelled by the rocket is
Because the area of the rectangles is (a) definitely smaller than the real area, and (b) the largest area that is definitely smaller, it is the greatest lower bound for the real area. The real area must be bigger than this. instead of a vague “quite good”, we have a definite statement.
SLIDE 51
Least upper bound
Velocity-time graph of a rocket
The diagram shows how to find a least overestimate of the area under a curve. This overestimate is called the least upper bound for the distance travel by the rocket in the first 60 s. This is because it is (a) definitely larger than the real distance, and (b) the least distance that is definitely larger.
Complete the diagram and find this estimate.
SLIDE 52
The least upper bound of the distance travelled by the rocket is
SLIDE 53
Bounded estimate
Greatest lower bound Least upper bound
The greatest lower bound for the distance travelled by the rocket in the first 60 s is 1630 m. The least upper bound for the distance travelled by the rocket is 2930 m.
Using s to stand for the distance travelled by the rocket in the first 60 s into a single inequality. Explain why this means that the estimate is now bounded. How could we narrow the bounds – that is, make the interval of the bound smaller?
SLIDE 54
Greatest lower bound Least upper bound
where d is the distance travelled by the rocket in the first 60 s.
This means that the estimate is now bounded between a least and a greatest value. We can narrow the bound, by increasing the number of rectangles being used, which is the same as making the width of the rectangles smaller.
The width of the rectangles above is the same, but the method is not changed if we make the width of the rectangles different, so long as we do not count an area twice.
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