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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?

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What do we understand about 'acceleration'?

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acceleration (m s –2 ) is

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

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v = u + at

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

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when acceleration is constant (uniform) and motion is in a straight line

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

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Describe the motion of this object

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How can we determine the displacement of the object?

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Area under the graph = 1500 + 4500 = 6000 m Area 1 = 5 × 300 = 1500 m Area 2 = ½bh = ½ × (35 –- 5) × 300 = 4500 m

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Area 1 = ut Area 2 = ½bh = ½ × (v – u) × t Since v = u + at and v – u = at vuvu t

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Area 1 = ut Area 2 = ½bh = ½ × (v – u) × t = ½ × at × t = ½ × at 2 vuvu t Area under the graph = displacement s

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Area 1 = ut Area 2 = ½bh = ½ × (v – u) × t = ½ × at × t = ½ × at 2 vuvu t s = ut + ½at 2

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v = u + at Start with Equation 1

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v 2 = (u + at) 2 and square it

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v 2 = u 2 + 2uat + a 2 t 2 v 2 = u 2 + 2a(ut+ ½at 2 )

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v 2 = u 2 + 2as

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What do we need to think about when using the equations of motion?

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What do the following quantities have in common? velocitydisplacement acceleration

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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.

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The sign convention Normally we use the following convention: positive + negative – positive + negative –

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

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v = u + at What does a positive value of acceleration mean?

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Using the normal sign convention –ve +ve

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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. –ve+ve

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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? –ve+ve

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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. –ve+ve

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Her acceleration is a = –1.19 m s –2. Notice that the acceleration has a negative value. Explain this. –ve+ve

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Now consider Christine running in the opposite direction. Notice that the sign convention remains the same. –ve+ve

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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. –ve+ve

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

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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. –ve+ve

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Her acceleration is a = 1.19 m s –2. Notice that the acceleration has a negative value. Explain this. –ve+ve

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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 0 m s –1 Final velocityAcceleration +10.49 m s –1

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

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

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OR A negative value means the object is losing speed while moving in the positive direction. –ve +ve Initial velocity 0 m s –1 Final velocityAcceleration +10.49 m s –1

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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. –ve+ve

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Step 1: Write down the sign convention. Step 2: Write down what you know (think suvat). sdisplacement uinitial velocity vfinal velocity aacceleration ttime Step 3: Any other information, eg acceleration due to force of gravity?

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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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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. –ve+ve

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

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

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

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

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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.

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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?

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Everyday acceleration Do you experience accelerations only in the horizontal?

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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.

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A stationary tennis ball Describe its motion.

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A tennis ball travelling vertically upwards 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. Describe its motion.

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A tennis ball dropped from a height 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. Describe its motion.

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Two tennis balls dropped from a height Predict the motion. Observe. Explain!

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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?

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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?

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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.

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© Erich Schrempp / Science Photo Library

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© Nicola Jones

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Suppose we can switch off air resistance. Which will hit the ground first? The elephant and the feather in free-fall

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The force of gravity near the Earths surface gives all objects the same acceleration. So why doesnt the feather reach the ground at the same time as the elephant? The elephant and the feather with air resistance

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We commonly use a negative to indicate downward motion. Dropping an elephant… 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.

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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 (m s –1 ) Distance fallen (m)Displacement (m) 00000 1 2 3 4 5

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

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

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Time (s) Speed (m s –1 ) Displacement (m) Velocity (m s –1 ) Distance ( m) Time (s) 0 0 0 0

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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. http://www.helpmyphys ics.co.uk/bouncing- ball.html

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Describe the motion. 0

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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! 0

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0 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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What assumptions are you making? How could your calculation be improved? Calculate the initial acceleration of a jumping popper.

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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.

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

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