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**In this section we will…**

…develop our understanding of using numbers and equations to describe motion.

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**What equations to describe motion have you come across before?**

Think and Talk What equations to describe motion have you come across before?

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**What do we understand about 'acceleration'?**

Think and Talk What do we understand about 'acceleration'?

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acceleration (m s–2) is

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**Rearrange this to give v =…**

rate of change of velocity per unit time Rearrange this to give v =…

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v = u + at

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**The Equations of Motion**

There are three equations which together are known as the equations of motion.

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**The Equations of Motion?**

When can I use… The Equations of Motion? when acceleration is constant (uniform) and motion is in a straight line

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**The Equations of Motion?**

Do I have to learn… The Equations of Motion? You need to be able to: select the correct formula identify symbols and units carry out calculations to solve the problems of real-life motion.

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**The Equations of Motion?**

Do I have to learn… The Equations of Motion? You need to be able to: carry out experiments to verify the equations of motion

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**To do this fully, you might find it an interesting challenge to…**

understand where the equations come from.

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**The Equations of Motion**

v = u + at Label the formula using correct symbols and units

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Equation 2 Describe the motion of this object

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Equation 2 How can we determine the displacement of the object?

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**Equation 2 Area under the graph = 1500 + 4500 = 6000 m Area 2 = ½bh**

= ½ × (35 –- 5) × 300 = 4500 m Area 1 = 5 × 300 = 1500 m

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**Equation 2 v Area 2 = ½bh = ½ × (v – u) × t Since v = u + at**

and v – u = at Area 1 = ut t

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**Equation 2 Area under the graph = displacement s v Area 2 = ½bh**

= ½ × (v – u) × t = ½ × at × t = ½ × at2 Area 1 = ut t

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**Equation 2 s = ut + ½at2 v Area 2 = ½bh = ½ × (v – u) × t = ½ × at × t**

Area 1 = ut t

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Equation 3 Start with Equation 1 v = u + at

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Equation 3 and square it v2 = (u + at)2

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**v2 = u2 + 2uat + a2t2 v2 = u2 + 2a(ut+ ½at2) **

Equation 3 v2 = u2 + 2uat + a2t2 v2 = u2 + 2a(ut+ ½at2) Equation 2!

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Equation 3 v2 = u2 + 2as

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**Using the Equations of Motion**

What do we need to think about when using the equations of motion? Symbols, units. Standard techniques for layout. Vector quantities and direction.

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**velocity displacement**

Using the Equations of Motion What do the following quantities have in common? velocity displacement acceleration Vector quantities and therefore positive direction must be chosen and the reverse direction given a negative sign.

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**The sign convention When dealing with vector quantities we**

must have both magnitude and direction. When dealing with one-dimensional kinematics (ie motion in straight lines) we use + and – to indicate travel in opposite directions. L

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**The sign convention Normally we use the following convention:**

negative – positive + negative – positive +

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**Take care – in some questions the sign convention is reversed**

Normally we use the following convention negative – positive + negative – positive + Take care – in some questions the sign convention is reversed

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**v = u + at Equation 1 and the Sign Convention**

What does a positive value of acceleration mean?

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**Using the normal sign convention**

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**–ve +ve Christine Arron is a 100-m sprint athlete.**

Immediately the starting pistol is fired, Christine accelerates uniformly from rest, reaching maximum velocity at the 50-m mark in 4.16 s. Her maximum velocity is 10.49 m s–1. Calculate her acceleration over the first 50 m of the race, showing full working. Christine Arron is a track and field sprint athlete, who competes for France. Statistics on elite athletic performance quoted from v = u + at 10.49 = 0 + a(4.16) a = 2.52 m s–2

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**–ve +ve Her acceleration is 2.52 m s–2.**

In this case, acceleration is a rate of change of velocity with time, with which we are familiar. A positive value means, in this case, increasing velocity with time. What else might it mean? Christine Arron is a track and field sprint athlete, who competes for France. Statistics on elite athletic performance quoted from v = u + at 10.49 = 0 + a(4.16) a = 2.52 m s–2

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**–ve +ve As she passes the finish line, Christine begins to slow down.**

She comes to rest in 8.20 s from a velocity of 9.73 m s–1. Calculate her acceleration, showing full working. Statistics on elite athletic performance quoted from v = u + at 0 = a(8.20) a = –1.19 m s–2

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**–ve +ve Her acceleration is a = –1.19 m s–2.**

Notice that the acceleration has a negative value. Explain this. Acceleration with a negative value: in this case losing speed in the positive direction.

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**–ve +ve Now consider Christine running in the opposite direction.**

Notice that the sign convention remains the same. Christine Arron is a track and field sprint athlete, who competes for France. Statistics on elite athletic performance quoted from v = u + at –10.49 = 0 + a(4.16) a = –2.52 m s–2

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–ve +ve Immediately the starting pistol is fired, Christine accelerates uniformly from rest, reaching maximum velocity at the 50-m mark in 4.16 s. Her maximum velocity is –10.49 m s–1 (why is it negative?). Calculate her acceleration over the first 50 m of the race, showing full working. Christine Arron is a track and field sprint athlete, who competes for France. Statistics on elite athletic performance quoted from v = u + at –10.49 = 0 + a(4.16) a = –2.52 m s–2

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**–ve +ve Her acceleration is –2.52 m s–2. What does the negative mean?**

Acceleration with a negative value: in this case gaining speed in the negative direction.

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**–ve +ve As she passes the finish line, Christine begins to slow down.**

She comes to rest in 8.20 s from a velocity of –9.73 m s–1. Calculate her acceleration, showing full working. Statistics on elite athletic performance quoted from v = u + at 0 = – a(8.20) a = 1.19 m s–2

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**–ve +ve Her acceleration is a = 1.19 m s–2.**

Notice that the acceleration has a negative value. Explain this. Acceleration with a positive value: in this case losing speed in the negative direction.

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**Equation 1 and the sign convention**

A positive value means gaining speed while moving in the positive direction. –ve +ve Initial velocity Final velocity Acceleration 0 m s–1 m s–1

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OR A positive value means the object is losing speed while moving in the negative direction. –ve +ve Initial velocity Final velocity Acceleration –9.73 m s–1 0 m s–1

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In summary: A negative value means the object is gaining speed while moving in the negative direction. –ve +ve Initial velocity Final velocity Acceleration –10.49 m s–1 0 m s–1

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OR A negative value means the object is losing speed while moving in the positive direction. –ve +ve Initial velocity Final velocity Acceleration 0 m s–1 m s–1

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–ve +ve Usain Bolt is a Jamaican sprinter and a three-times Olympic gold medallist. Immediately the starting pistol is fired, Usain accelerates uniformly from rest. He reaches 8.70 m s–1 in 1.75 s. Calculate his displacement in this time.

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**Using the Equations of Motion**

Step 1: Write down the sign convention. Step 2: Write down what you know (think suvat). s displacement u initial velocity v final velocity a acceleration t time Step 3: Any other information, eg acceleration due to force of gravity?

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**Using the Equations of Motion**

Step 4: Select formula – use data sheet. Step 5: Substitute values then rearrange formula. Step 6: Write the answer clearly, including magnitude and direction, and units.

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–ve +ve Usain Bolt is a Jamaican sprinter and a three-times Olympic gold medallist. Immediately the starting pistol is fired, Usain accelerates uniformly from rest. He reaches 8.70 m s-1 in 1.75 s. Calculate his displacement in this time.

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–ve +ve s = ? m u = 0 m s–1 v = 8.70 m s–1 a = ? t = 1.75 s

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-ve +ve s = ? m u = 0 m s–1 v = 8.70 m s–1 a = ? t = 1.75 s

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-ve +ve s = ? m u = 0 m s–1 v = 8.70 m s–1 a = ? t = 1.75 s

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-ve +ve s = ? m u = 0 m s–1 v = 8.70 m s–1 a = ? t = 1.75 s

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**In the previous section we developed…**

…our understanding of using graphs to describe motion …our skills in interpreting graphs of motion …our skills in describing motion using physics terms correctly. Written to follow on from ODU 1c).

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**In this section we planned to…**

…develop our understanding of using numbers and equations to describe motion.

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**Next we will bring all of this together and use…**

…our understanding of using graphs to describe motion …our skills in interpreting graphs of motion …our skills in describing motion using physics terms correctly … our understanding of using numbers and equations to describe motion for vertical motion

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**Everyday acceleration**

What sort of accelerations do you experience in everyday life? How can this be investigated? Using Vernier Wireless Accelerometer and Logger Pro to investigate acceleration when walking, jumping, jogging etc. Images from Microsoft Clip Art.

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**Everyday acceleration**

Do you experience accelerations only in the horizontal? Using Vernier Wireless Accelerometer and Logger Pro to investigate acceleration when walking, jumping, jogging etc. Images from Microsoft Clip Art.

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**Everyday acceleration**

An accelerometer (a device which measures acceleration in three dimensions) can be used to investigate accelerations which you experience in everyday life. Using Vernier Wireless Accelerometer and Logger Pro to investigate acceleration when walking, jumping, jogging etc. Images from Microsoft Clip Art.

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**A stationary tennis ball**

Holding the ball stationary, describe its motion. Images from Microsoft Clip Art. Describe its motion.

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**A tennis ball travelling vertically upwards**

Describe its motion. Film the ball as it is thrown upwards and use tracker.jar to analyse its motion. Once you have done this, describe the motion in detail using the words velocity, acceleration and displacement. Link to tracker.jar tutorial lesson. Images from Microsoft Clip Art.

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**A tennis ball dropped from a height**

Describe its motion. Film the ball as it falls and use tracker.jar to analyse its motion. Once you have done this, describe the motion in detail using the words velocity, acceleration and displacement. Link to tracker.jar tutorial lesson. Images from Microsoft Clip Art.

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**Two tennis balls dropped from a height**

Predict the motion. Observe. Explain! The two balls appear to be identical. Both drop and hit the ground at the same time. A student is asked to retrieve them and will hopefully observe a significantly different mass. One ball is as bought, the other has been filled with liquid. However, both hit the ground at exactly the same time. Once the students are aware of the difference in mass they may try to observe a difference between the times at which the balls hit the ground in order to fit this in with a common misconception that heavier objects fall more quickly than lighter objects. Images from Microsoft Clip Art.

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**Two tennis balls dropped from a height**

Was your initial prediction that the two identical tennis balls dropped from the same height would hit the ground at exactly the same time? The two balls appear to be identical. Both drop and hit the ground at the same time. A student is asked to retrieve them and will hopefully observe a significantly different mass. One ball is as bought, the other has been filled with liquid. However, both hit the ground at exactly the same time. Once the students are aware of the difference in mass they may try to observe a difference between the times at which the balls hit the ground in order to fit this in with a common misconception that heavier objects fall more quickly than lighter objects. Images from Microsoft Clip Art.

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**Two tennis balls dropped from a height**

What did you think when you discovered that one ball had a significantly greater mass than the other? What do you think should have happened? What did you observe? The two balls appear to be identical. Both drop and hit the ground at the same time. A student is asked to retrieve them and will hopefully observe a significantly different mass. One ball is as bought, the other has been filled with liquid. However, both hit the ground at exactly the same time. Once the students are aware of the difference in mass they may try to observe a difference between the times at which the balls hit the ground in order to fit this in with a common misconception that heavier objects fall more quickly than lighter objects. Images from Microsoft Clip Art.

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**The mass does not matter! **

Both balls will hit the ground at the same time when dropped from the same height. If you do not believe this, tracker.jar will allow you to analyse the motion. The two balls appear to be identical. Both drop and hit the ground at the same time. A student is asked to retrieve them and will hopefully observe a significantly different mass. One ball is as bought, the other has been filled with liquid. However, both hit the ground at exactly the same time. Once the students are aware of the difference in mass they may try to observe a difference between the times at which the balls hit the ground in order to fit this in with a common misconception that heavier objects fall more quickly than lighter objects. Images from Microsoft Clip Art.

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**© Erich Schrempp / Science Photo Library**

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© Nicola Jones

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**The elephant and the feather in free-fall**

Suppose we can switch off air resistance. Which will hit the ground first? Images from Clipart

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**The elephant and the feather with air resistance**

The force of gravity near the Earth’s surface gives all objects the same acceleration. So why doesn’t the feather reach the ground at the same time as the elephant? Images from Clipart

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Dropping an elephant… We commonly use a negative to indicate downward motion. Images from Clipart but be warned – you may come across questions in which the sign convention is reversed.

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**Dropping an elephant in the absence of air resistance.**

Images from Clipart

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**Dropping an elephant in the absence of air resistance**

Calculate speed, velocity, distance and displacement at 1-s intervals. Time (s) Speed (m s–1) Velocity Distance fallen (m) Displacement (m) 1 2 3 4 5 Print off to save time!!!

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Sketch graphs to show how the speed, distance, velocity and displacement vary with time during the free-fall. Speed Distance Time Time Velocity Displacement Time Time

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**What is missing from the graph sketches?**

Sketch graphs to show how the speed, distance, velocity and displacement vary with time during free-fall. What is missing from the graph sketches? Speed Distance Time Time Velocity Displacement Time Time

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Speed (m s–1) Distance ( m) Time (s) Time (s) Velocity (m s–1) Displacement (m) Time (s) Time (s)

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**A tennis ball dropped from a height and allowed to bounce**

Consider the ball being dropped, allowed to bounce and return to its original height. Sketch your predictions for speed–time, velocity–time and acceleration–time graphs.

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**Compare to the results from this simulation.**

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Describe the motion.

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**When dropped, the ball gains speed in the negative direction hence the –ve sign for acceleration.**

The ball then loses speed in the positive direction, coming to rest at the original height. Does this happen in real life? Explain!

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a ( m s–2) Time (s)

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**Consider a tennis ball thrown upwards and allowed to fall back to its starting position.**

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**A tennis ball thrown upwards then allowed to fall back to its starting position**

Sketch the velocity, speed and acceleration graphs to describe its motion until it returns to its starting position.

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**Virtual Higher Experiments → Higher Physics → Mechanics and Properties of Matter → Activity 5b.**

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**What is the force acting on a tennis ball thrown upwards?**

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**Estimate the initial acceleration of a jumping popper.**

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**Calculate the initial acceleration of a jumping popper.**

What assumptions are you making? How could your calculation be improved?

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**Observe what happens when the groaning tube is dropped.**

Explain!

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**In the previous section we developed…**

…our understanding of using graphs to describe motion …our skills in interpreting graphs of motion …our skills in describing motion using physics terms correctly. Written to follow on from ODU 1c).

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**Review your progress! In this section we…**

…developed our understanding of using numbers and equations to describe motion. Review your progress!

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