Measuring Motion - St Leonard's College

Year 10 Physics

Measuring Motion

Name: _________________

Motion Formulas

Speed = distance Velocity = displacement v = dt

time time

Acceleration = change in velocity a = ∆vt

= v-u

t

time

Equation of motion: v = u + at

Measuring Motion Activity

Distance is a measure of how far an object has moved. Displacement is a measure of how far an object has moved in a given direction. Speed is a measure of how far an object has moved in a given time.

i.e speed = distance time Velocity is a measure of how far an object has gone in a particular direction in a given time.

i.e Velocity = displacement time Instructions

1. Walk at a steady pace around the basketball court following the lines and turning at soon as you can.

2. Using a stopwatch, beanbags and a tape measure, record the distance and displacement that you walk at five second intervals around the court until you’ve used all your beanbags.

Results

Time (s) Total Distance (m) Displacement (m)

0

5

10

15

20

25

30

Analysis

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Determine the average speed and the magnitude of the average velocity for each time interval. Show your calculations for full marks.

Time interval (s)

Average Speed (m/s)

Average velocity (m/s)

0

5

10

15

20

25

30

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Graphs

On the grids below create the following two graphs. Include all standard graph features.

Graph 1: Total distance versus time

Graph 2: Displacement versus time

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Questions

1. Measure the gradient of the distance time graph. (a) Show your calculations and record your answer with units.

What can you say about the gradient of this graph when compared to the average speed?

(a) When finding the velocity of an object, what else should have been recorded apart from the magnitude?

(b) What could you have done to include this information?

Describe some limitations of the experiment and suggest how it could be improved.

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Introduction Determining the Equation for Speed A car travelling at 60 kilometres per hour (km/h) covers 60 kilometres in one hour. A car travelling at 30 km/h covers 30 kilometres in one hour.

1. How far would a car travelling at 50 km/h cover in one hour?

2. How far would a car travelling at 50 km/h travel in two hours?

3. How far would a car travelling at 50 km/h travel in 30 minutes?

4. Based on your answers above write out a worded equation that gives the distance travelled by a car in terms of speed and time.

A car that travels 60 kilometres in one hour has an average speed of 60 kilometres per hour (km/h). A car that travels 30 kilometres in one hour has an average speed of 30 km/h.

5. A car travels 100 km in one hour. What it its average speed?

6. A car travels 50 km in half an hour. What is its average speed?

7. A car travels 300 km in 3 hours. What is its average speed?

8. Based on your answers above write out a worded equation that gives the average speed of a car in terms of distance and time.

Questions to think about

Why do you feel pushed to the left when the bus you are in turns to the right?

Why does it hurt when you catch a fast-moving ball with your bare hands?

Why can’t a tennis ball bounce higher than the height from which it was dropped?

Why are cars deliberately designed to crumple in road crashes?

Who would win a race between a sea turtle, a dolphin and an Olympic swimmer? Your questions

1.

2.

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Speed Speed is a measure of the rate at which an object moves over a distance. Speed tells us how fast a thing is moving. For example, the speed of a car tells us how fast it is moving by telling us how far it will move in a set time. If according to the speedo (speedometer) a car (1) is travelling at 100 km/h, then if it remains at this speed then it should travel 100 km every hour. If another car (2) was travelling more slowly at 80 km/h, then at this speed, the car travels 80 km every hour. Consider the following table. Table 1

Car Speed Time Distance travelled

1 100 km/h ½ h 50 km

1 100 km/h 1 h 100 km

1 100 km/h 2h 200 km

2 80 km/h ½ h 40 km

2 80 km/h 1h 80 km

2 80 km/h 2h 160 km

We can say that; Example 1 What is the distance travelled by a train in 2.5 hours if it is moving at 70 km/h? Solution Step 1 Write out the formula d = v x t Step 2 Substitute the correct values d = 70 x 2.5 Step 3 Calculate d = 175

Step 4 Make sure you include the correct units d = 175 km. Questions 1. A sports car travels at its top speed of 200 km/h. How far will it travel in a) 5 hours b) 30 minutes

Distance = speed x time

d = vt

(d = distance, v = speed, t = time)

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Average Speed Speed is also useful because we can use it to work out how long a journey will take. Table 2

Car Length of journey (distance)

Speed (average)

Time taken

1 200 km 100 km/h 2 h

1 350 km 100 km/h 3½ h

2 40 km 80 km/h ½ h

2 200 km 80 km/h 2½ h

The most conventional way of summarising this is to use;

average speed = distance travelled

time taken v =

d

t

Units of speed The speed of vehicles is usually expressed in kilometres per hour (km/h or kph). However, sometimes it is more convenient to express speed in the units of metres per second (m/s). Speed is always expressed as a unit of distance divided by a unit of time. Example 2 What is the average speed of an aeroplane that travels from Perth to Melbourne, a distance of 2730 km by air, in 3 hours? Solution

Step 1 Write out the formula v = d

t

Step 2 Substitute the correct values v = 2730

3

Step 3 Calculate v = 910

Step 4 Make sure you include the correct units v = 910 km/h.

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Converting Units If an answer is required in different units it is generally easier to convert the units before completing any calculations. Use the following conversions: 1km = 1000 m 1 hour = 60 minutes 1 minute = 60 seconds For example, if you need to convert hours into seconds multiply by 60 to get minutes, then multiply by 60 again to get seconds. How do you know whether to multiply or divide? Use a logic check. There will be many seconds in one hour, so it makes sense to multiply. You can also check that the units cancel out. Example 2 (continued) What is the average speed express the speed in m/s? Step 1 Convert the units d = 2730km x 1000m/km d = 2 730 000 m t = 3h x 60min/h x 60s/min t = 10 800 s

Step 2 Write out the formula v = d

t

Step 3 Substitute the converted values v = 2 730 000

10800

Step 4 Calculate v = 253

Step 5 Make sure you include the correct units v = 253 m/s. A very useful conversion factor is: Questions 1. In an Olympic swimming competition, an electronic clock starts as the starter’s gun is

fired. The clock stops when the swimmer touches the pad at the end of the pool. Work out the average speed for this swimmer - Backstroke - 100 m in 60 s.

a) in m/s

b) in km/h (by converting the metres to km and the seconds to hours)

c) in km/h (using the conversion in the box above)

x 3.6

m/s km/h

÷ 3.6

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Motion Key Terms – Distance v Displacement, Speed v Velocity

Scalars: A scalar quantity has size (magnitude) only. It has no direction.

Eg. distance, speed, mass and time are all scalar quantities.

Vectors: A vector quantity has size (magnitude) and a direction.

Eg. displacement, velocity, acceleration and weight are all vector quantities.

SI Units: Standard International Units.

Eg. metres, kilograms, seconds, Newton’s,

joules

Distance: How far an object has moved.

SI Units: metres (m).

Displacement: How far an object is from its

original position.

SI Units: metres (m) + direction.

Speed: The rate at which distance changes over time. How far an object has moved

in a given time period.

SI Units: metres/second (m/s).

Velocity: The rate at which displacement changes over time. How far an object has

moved in a given time relative to its original position, in a particular direction.

SI Units: metres/second (m/s) + direction.

speed = distance

time

v = d/t

velocity = displacement

time

v = d/t

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Ready, Set, Go - Questions!

1. Write the formula used to calculate average speed in symbols and state which quantity each

symbol represents.

2. Explain the difference between speed and velocity. Use an example to support your explanation.

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3. Determine the average speed of each of the following

a. a racehorse that wins the 3200 m Melbourne Cup in a time of 3 min 20 s (in m/s)

b. a kangaroo fleeing from a dingo, which bounds a distance of 2.5 km in 3 min (in m/s)

c. a dolphin that just manages to keep up with a speeding boat for a distance of 2 km for a period of 3 min (km/h)

d. a sea turtle that is able to maintain its maximum speed for 0.5 h. In that time it can swim a distance of 16 km (km/h)

e. An Olympic swimmer who completes a 1500 m training swim in 15 min (km/h)

f. A mosquito that flies a distance of 2 m in 4 s (cm/s)

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4. Use your answers to question 3 to answer the following questions a. Suggest if a kangaroo could win the Melbourne Cup?

b. Who would win a race between a sea turtle, a dolphin and Olympic swimmer?

5. How long would it take you to walk from Melbourne to Sydney, a distance of 900 km, if you walked an average speed of:

a. 5 km/h without stopping?

b. 5km/h for 10 h each day?

c. 1.5 m/s without stopping?

6. How far can a snail crawl if it moves at an average aped of 8.0 cm/min for: a. 3 min?

b. 3 h?

7. A swimmer swims the 100 m breast stroke event in 68 s. The event is completed in a pool that is 50 m long. She finishes the event at the same end of the pool from which she started. If she begins the event by swimming due north, and takes 35 s to swim the first 50 m, calculate her:

a. average speed for the whole event

b. average velocity for the first 50 m

c. average velocity for the whole event

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8. A swimmer completes a 1500 m race in 870 s. Calculate the swimmers average speed in; a. m/s

b. km/h

Distance, displacement, speed and velocity questions Is the following a Vector or Scalar? Velocity - Speed - Distance - Displacement – 1. Q: A car moved 80 km to the South. What is its displacement? A: 20 km South B: 50 km East C: 80 km South D: 160 km North

------------------------------------- 2. Q: A car moved 60 km East and 90 km West. What is the distance? A: 30 km B: 60 km C: 90 km D: 150 km

------------------------------------- 3. Q: A car moved 60 km East and 90 km West. What is the displacement? A: 30 km West B: 60 km West C: 30 km East D: 150 km

------------------------------------- 6. Q: What is the average velocity of a car that moved 40 km East and 80 km West in 2 hours? A: 5 km/h West B: 10 km/h West C: 15 km/h West D: 20 km/h West ------------------------------------- 8. Q: How far will a car travel in 2 hours at 20 m/s? A: 144 km B: 158 km C: 168 km D: 234 km

EXT 9. Q: If car A is at 40 km/h and car B is at 10 km/h in the opposite direction, what is the velocity of the car A relative to the car B? A: 10 km/h B: 20 km/h C: 40 km/h D: 50 km/h

------------------------------------- EXT 10. Q: If you are walking at constant velocity of 8 km/h and a car passed you by at the speed of 30 km/h from behind, what is the car's velocity from your viewpoint? A: 22 km/h B: 30 km/h C: 38 km/h D: 40 km/h

------------------------------------- EXT 11. Q: If car A is at 70 km/h and car B is at 50 km/h in the same direction, what is the velocity of the car A relative to the car B? A: 10 km/h B: 20 km/h C: 30 km/h D: 40 km/h

------------------------------------- EXT 12. Q: If a car moves 12 km North, 19 km East, and 12 km South, what is its displacement? A: 12 km East B: 19 km East C: 31 km East D: 43 km East

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Graphing motion Displacement-time graphs

Displacement-time graphs show how the position of a moving object changes over time.

The position at any time can be determined by reading directly from the graph.

The gradient of a displacement-time graph tells us the velocity.

i.e. velocity = rise run = y2 – y1 x2 – x1

Velocity-time graphs

Velocity-time graphs measure how fast an object is moving at any particular moment in time.

The velocity can be determined by reading directly from the graph.

The acceleration can be found by measuring the gradient.

The displacement can be found by measuring the area under the curve. Displacement-Time Graph

Velocity-Time Graph Note: This velocity graph corresponds (approximately) to the displacement time graph above. The dotted lines show sudden changes in velocity – they appear to be instantaneous in the displacement-time graph, but more realistically they should be lines with steep gradients as shown here.

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Plotting motion graphs The table below shows the motion of a car starting from rest and moving along a road.

Time (s) 0 5 10 15 20 25 30 35 40

Position (m) 0 0 10 20 30 30 30 25 20

1. Plot a displacement-time graph showing the motion of the car. Include a title and

all relevant graph features.

2. Describe what the car is doing

between:

(a) 0-5 seconds

(b) 5-20 seconds

(c) 20-30 seconds

(d) 30-40 seconds

3. Calculate the velocity of the car between:

(a) 0-5 seconds

(b) 5-20 seconds

(c) 20-30 seconds

(d) 30-40 seconds

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4. Plot a velocity-time graph of the motion of the car. Include a title and all relevant graph features.

5. What is the total distance travelled by the car?

6. What is the displacement of the car at the end of the journey?

7. What is the average speed of the car (over the entire journey)?

8. What is the average velocity of the car (over the entire journey)?

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Position-Time Graph to Velocity-Time Graph Practice

1. Create the corresponding velocity-time graph for either position graph

Determine the velocity during each different stage of motion and write it on the graph.

Determine the maximum and minimum velocity.

Sketch axes for your velocity-time graph directly below each position-time graph with a time scale that lines up. Ensure that the limits of your y-axis are the maximum and minimum velocity determined previously.

Sketch the velocity-time graph corresponding to the position-time graph. 2. Determine the total distance travelled and final displacement 3. Determine the average speed and average velocity for either graph.

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Interpreting motion graphs Entrée task: which of the graphs describes an object with a constant speed? Main task: describe the motion of the object in each of the following graphs. Displacement-Time Graphs:

Velocity-Time Graphs:

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Measuring speed

When Formula One racing driver Sebastian Vettel broke the Australian Grand Prix lap record in 2011, he completed a 5.303 km lap in 83.529 seconds. His average speed was

Step 1 v = d

t

Step 2 v = 5.303´1 000

83.529

Step 3 and 4 = 63.49 m/s To convert from m/s to km/h multiply by 3.6

63.49 x 3.6 ~ 229 km/h However, he was able to speed down the straight at speeds of up to 320 km/h. Clearly, the average speed does not provide much information about the speed at any particular instant during the race. Speedometers The speedometer inside a vehicle has a pointer that rotates further to the right as the wheels of the car turn faster. It provides a measure of the instantaneous speed. Ticker Timers

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Measuring speed 1. Explain the difference between instantaneous speed and average speed. Use an

example to support your explanation.

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………………………………………………………………………………………………………………………………………………….

………………………………………………………………………………………………………………………………………………….

2. Make a list of reasons why a speedometer reading might not be accurate. Include in

your list anything that could change the diameter of the vehicles tyres.

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………………………………………………………………………………………………………………………………………………….

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Changing speed - Acceleration The accelerator of a car is given that name because pushing down on it usually makes the car accelerate. When an object moves in a straight line its acceleration is a measure of the rate at which it changes speed. Acceleration tells you how quickly the speed changes. The average acceleration can be calculated by dividing the change in speed by the time taken for the change. That is:

acceleration = change in speed

time taken We can write this is the form of an equation: Acceleration is normally given the symbol a, and it has the standard international units m/s2. If the change in speed is an increase, the acceleration is positive. If the change in speed is a decrease, the acceleration is negative and called deceleration.

Change in velocity = Dv = v - u

a = Dv

t=

v - u

t

a = acceleration v = final velocity u = initial velocity

∆ (delta, a capital D in the Greek alphabet) represents “change in’

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Example A car travelling at 60 km/h increases its speed to 100 km/h in 5.0 seconds. What is its average acceleration?

Step 1 a = v

t

Step 2 a = 100 - 60

5

Step 3 and 4 = 8 km/h per sec. This is on average, the car increasing its speed by 8 km/h each second. Questions

1. Determine the change in speed of each of the following situations: a. The driver of a car heading along a freeway at 100 km/h slows down to 60 km/h

as the traffic gets heavier.

b. A fielder catches a cricket ball travelling towards him at 20 m/s Drag racing The sport of drag racing is a test of acceleration. From a standing start, cars need to cover a distance of 400metres in the shortest possible time. To do this they need to reach high speeds very quickly. The fastest drag racing cars can reach speeds of more than 500 km/h in less than 5.0 seconds. Example The average acceleration of a drag racer that reaches a speed of 506 km/h in 4.6 seconds is:

Step 1 average acceleration = Dv

t

Step 2 a = 506 - 0

4.6

Step 3 and 4 = 110 km/h per sec. This is on average, the car increasing its speed by 110 km/h each second. Acceleration can also be expressed in m/s/s (that is metres per second per second) or m/s2 (that is metres per second squared). A change in speed of 506 km/h is also 141 m/s (divide by 3.6). The average acceleration of the drag racer is;

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t

Step 2 a = 141- 0

4.6

Step 3 and 4 = 31 m/s2.

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Slowing down Once the drag racer has completed the required distance of 400 metres, it needs to stop before it reaches the end of the track. The fastest cars can release parachutes so that they can stop in time. The acceleration of a car that comes to rest in 5.4 seconds from a speed of 506 km/h is:


t

Step 2 a = 0 - 056

5.4

Step 3 and 4 = -93.7 km/h per sec. This negative acceleration can be expressed as a deceleration of 93.7 km/h/s Questions

1. Explain the difference between acceleration and deceleration.

2. Determine the acceleration of a car that goes from rest to 60 km/h in 20 seconds.

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Determining the final velocity

The final velocity after a certain period of time can be determined by rearranging the equation for acceleration to give: Questions

1. A car that was travelling at 40 km/h accelerates for 10 seconds with an average acceleration of 1 km/h/s. What is its final speed?

2. A car that was travelling at 60km/h decelerates for 20 seconds at an average deceleration of 3 km/h/s. What is its final speed?

3. A particle has a velocity of 2.0 m/s. Ten seconds later, it has a velocity of 8.0 m/s. What was the average acceleration of the particle?

Next: Select just one of the following acceleration worksheets complete.

v = u + at

a = acceleration v = final velocity u = initial velocity

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ACCELERATION QUESTIONS – Moderate (choose only one sheet!)

Average acceleration = change in velocity a = Δv v = u + at time taken t

1. a) If a rocket accelerates from 50 m/s to 70 m/s in 10 seconds, what is its acceleration (in m/s2)?

b) A person starting from rest, accelerates at 0.5 m/s2 for 6 seconds. What is their final velocity?

c) A car starts its journey by accelerating for 4 seconds until it reaches a velocity of 8m/s. What was its acceleration (in m/s2)?

2. If a car accelerates from 50 km/hr to 70 km/hr in 5 seconds, what is

(a) the cars change in velocity in km/hr? (b) the cars acceleration in km/hr/s?

(c) the cars change in velocity in m/s

(d) the cars acceleration in m/s2?

3. A car accelerates from stationary to 60 km/hr with an acceleration of 12 km/hr/s. How long does this take?

4. A runner accelerates at 2km/hr/s for 4 seconds. a) What is her change in speed?

b) If she was originally walking slowly at 1 km/hr before this acceleration, what was her final speed?

5. (a) What is the acceleration of a car that goes from 100 km/hr to 20 km/hr in 8 seconds?

(b) What is another word for this acceleration?

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ACCELERATION QUESTIONS – Challenging (choose only one sheet!)

Average acceleration = change in velocity a = Δv v = u + at time taken t 1. a) If a rocket accelerates from 50 m/s to

70 m/s in 10 seconds, what is its

acceleration?

b) If a car accelerates from 50 km/hr to 70 km/hr in 5 seconds, what is its acceleration in m/s?

2) A car accelerates from stationary to 60 km/hr with an acceleration of 12 km/hr/s. How long does this take?

3) A runner accelerates at 2km/hr/s for 4 seconds.

a) What is her change in speed in m/s?

b) If she was originally walking slowly

at 1 km/hr before this acceleration, what was her final speed in m/s?

4) A car accelerates for 10 seconds at 5 km/h/s. If its final speed is 90 km/h what was its original speed?

5) (a) What is the acceleration of a car that goes from 100 km/hr to 20 km/hr in 8 seconds?

(b) What is another word for this acceleration?

Measuring Motion - St Leonard's College

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