Topic 2 Mechanics Use the syllabus and this REVISION POWERPOINT to aid your revision

No movement distance time

Constant speed distance time The gradient of this graph gives the speed

Constant speed distance time fast slow

Getting faster (accelerating) distance time

A car accelerating from stop and then hitting a wall distance time

Speed against time graphs speed time

No movement speed time

Constant speed speed time fast slow

Getting faster? (accelerating) (A straight line is “constant acceleration”) speed time a = v – u t (v= final speed, u = initial speed) v u The gradient of this graph gives the acceleration

Getting faster? (accelerating) speed time The area under the graph gives the distance travelled

A dog falling from a tall building (no air resistance) speed time Area = height of building

Displacement Displacement the distance moved in a stated direction (the distance and direction from the starting point). A VECTOR

Displacement/time graphs Usually in 1 dimension (+ = forward and - = backwards) Displacement/ m Time/s

Velocity? Velocity is the rate of change of displacement. Also a VECTOR

Velocity/time graphs Usually in 1 dimension (+ = forward and - = backwards) velocity/m.s -1 Time/s Ball being thrown into the air, gradient = constant = m.s -2

Acceleration? Acceleration is the rate of change of velocity. Also a VECTOR

An interesting example velocity We have constant speed but changing velocity. Of course a changing velocity means it must be accelerating!

Acceleration/time graphs Usually in 1 dimension (+ = up and - = down) accel/m.s -2 Time/s Acceleration = constant = m.s -2

Average speed/velocity? Average speed/velocity is change in distance/displacement divided by time taken over a period of time.

Instantaneous speed/velocity? Instantaneous speed/velocity is the change in distance/displacement divided by time at one particular time.

The equations of motion The equations of motion can be used when an object is accelerating at a steady rate There are four equations relating five quantities u initial velocity, v final velocity, s displacement, a acceleration, t time SUVAT equations

The four equations 1This is a re-arrangement of 2This says displacement = average velocity x time 3With zero acceleration, this becomes displacement = velocity x time 4Useful when you don’t know the time NOT in data book

Beware! All quantities are vectors (except time!). These equations are normally done in one dimension, so a negative result means displacement/velocity/acceleration in the opposite direction.

Example 1 Mr Rayner is driving his car, when suddenly the engine stops working! If he is travelling at 10 ms -1 and his decceleration is 2 ms -2 how long will it take for the car to come to rest? u = 10 ms -1 v = 0 ms -1 a = -2 ms -2 t = ? s v = u + at 0 = t 2t = 10 t = 5 seconds

Example 2 Jack steps into the road, 30 metres from where Mr Rayner’s engine stops working. Mr Rayner does not see Jack. Will the car stop in time to miss hitting Jack? u = 10 ms -1 v = 0 ms -1 a = -2 ms -2 t = 5 s s = ? m v 2 = u 2 + 2as 0 2 = x-2s 0 = s 4s = 100 s = 25m, the car does not hit Jack. 

Example 3 A ball is thrown upwards with a velocity of 24 m.s -1. When is the velocity of the ball 12 m.s -1 ? u = 24 m.s -1 a = -9.8 m.s -2 v = 12 m.s -1 v = u + at 12 = t -12 = -9.8t t = 12/9.8 = 1.2 seconds

Example 3 A ball is thrown upwards with a velocity of 24 m.s -1. When is the velocity of the ball -12 m.s -1 ? u = 24 m.s -1 a = -9.8 m.s -2 v = -12 m.s -1 v = u + at -12 = t -36 = -9.8t t = 36/9.8 = 3.7 seconds

Example 3 A ball is thrown upwards with a velocity of 24 m.s -1. What is the displacement of the ball at those times? (t = 1.2, 3.7) t = 1.2, v = 12, a = -9.8, u = 24 s = ? s = ut + ½at 2 = 24x1.2 + ½x-9.8x1.2 2 s = 28.8 – = 21.7 m

Example 3 A ball is thrown upwards with a velocity of 24 m.s -1. What is the displacement of the ball at those times? (t = 1.2, 3.7) t = 3.7, v = 12, a = -9.8, u = 24 s = ? s = ut + ½at 2 = 24x3.7 + ½x-9.8x3.7 2 s = 88.8 – = 21.7 m (the same?!)

Example 3 A ball is thrown upwards with a velocity of 24 m.s -1. What is the velocity of the ball 1.50 s after launch? u = 24, t = 1.50, a = -9.8, v = ? v = u + at v = x1.50 = 9.3 m.s -1

Gravity = air resistance Terminal velocity gravity As the dog falls faster and air resistance increases, eventually the air resistance becomes as big as (equal to) the force of gravity. The dog stops getting faster (accelerating) and falls at constant velocity. This velocity is called the terminal velocity. Air resistance

Falling without air resistance gravity The ball falls faster and faster and faster……. It gets faster by 9.81 m/s every second (9.81 m/s 2 ) This number is called “g”, the acceleration due to gravity.

Falling without air resistance? speed time Gradient = acceleration = 9.8 m.s -2

Velocity/time graphs Taking upwards are the positive direction velocity/m.s -1 Time/s Ball being thrown into the air, gradient = constant = m.s -2

Falling with air resistance? velocity time Terminal velocity

Gravity Gravity is a force between ALL objects! Gravity

The size of the force depends on the mass of the objects. The bigger they are, the bigger the force! Small attractive force Bigger attractive force

Gravity The size of the force also depends on the distance between the objects.

Mass and weight Mass is a measure of the amount of material an object is made of. It is measured in kilograms. Weight is the force of gravity on an object. It is measured in Newtons.

Calculating weight To calculate the weight of an object you multiply the object’s mass by the gravitational field strength wherever you are. Weight (N) = mass (kg) x gravitational field strength (N/kg)

Newton’s 1 st Law An object continues in uniform motion in a straight line or at rest unless a resultant external force acts Does this make sense?

Newton’s 2 nd law There is a mathematical relationship between the resultant force and acceleration. Resultant force (N) = mass (kg) x acceleration (m/s 2 ) F R = ma It’s physics, there’s always a mathematical relationship!

Newton’s 3 rd law If a body A exerts a force on body B, body B will exert an equal but opposite force on body A. Hand (body A) exerts force on table (body B) Table (body B) exerts force on hand (body A)

Free-body diagrams Shows the magnitude and direction of all forces acting on a single body The diagram shows the body only and the forces acting on it.

Examples Inclined slope W (weight) R (normal reaction force) F (friction) If a body touches another body there is a force of reaction or contact force. The force is perpendicular to the body exerting the force

Examples Mass hanging on a rope W (weight) T (tension in rope)

Momentum Momentum is a useful quantity to consider when thinking about "unstoppability". It is also useful when considering collisions and explosions. It is defined as Momentum (kg.m.s -1 ) = Mass (kg) x Velocity (m.s -1 ) p = mv

Law of conservation of momentum The law of conservation of linear momentum says that “in an isolated system, momentum remains constant”. We can use this to calculate what happens after a collision (and in fact during an “explosion”).

Conservation of momentum In a collision between two objects, momentum is conserved (total momentum stays the same). i.e. Total momentum before the collision = Total momentum after Momentum is not energy!

Momentum is a vector Momentum is a vector, so if velocities are in opposite directions we must take this into account in our calculations

An even harder example! Snoopy (mass 10kg) running at 4.5 m.s -1 jumps onto a skateboard of mass 4 kg travelling in the opposite direction at 7 m.s -1. What is the velocity of Snoopy and skateboard after Snoopy has jumped on? I love physics

An even harder example! 10kg 4kg-4.5 m.s -1 7 m.s -1 Because they are in opposite directions, we make one velocity negative 14kg v m.s -1 Momentum before = 10 x x 7 = = -17 Momentum after = 14v

An even harder example! Momentum before = Momentum after -17 = 14v V = -17/14 = m.s -1 The negative sign tells us that the velocity is from left to right (we choose this as our “negative direction”)

Impulse Ft = mv – mu The quantity Ft is called the impulse, and of course mv – mu is the change in momentum (v = final velocity and u = initial velocity) Impulse = Change in momentum

Impulse Ft = mv – mu F = Δp/Δt

Units Impulse is measured in N.s (Ft) or kg.m.s -1 (mv – mu)

Impulse Note; For a ball bouncing off a wall, don’t forget the initial and final velocity are in different directions, so you will have to make one of them negative. In this case mv – mu = -3m -5m = -8m 5 m/s -3 m/s

Another example A tennis ball (0.3 kg) hits a racquet at 3 m/s and rebounds in the opposite direction at 6 m/s. What impulse is given to the ball? Impulse = mv – mu = = 0.3x-6 – 0.3x3 = -2.7kg.m.s -1 3 m/s -6 m/s

Work In physics, work is the amount of energy transformed (changed) when a force moves (in the direction of the force)

Calculating work The amount of work done (measured in Joules) is equal to the force used (Newtons) multiplied by the distance the force has moved (metres). Force (N) Distance travelled (m)

Work = Fscosθ s F θ What if the force is at an angle to the distance moved?

Work done (J) = Force (N) x distance (m) A man lifts a mass of 120 kg to a height of 2.5m. How much work did he do? Force = weight = 1200N Work = F x d = 1200 x 2.5 Work = 3000 J

Power! Power is the amount of energy transformed (changed) per second. It is measured in Watts (1 Watt = 1 J/s) Power = Energy transformed time

Work done in stretching a spring F/N x/m Work done in strectching spring = area under graph

ΔE p = mgΔh Joules kg N/kg or m/s 2 m

Kinetic energy Kinetic energy of an object can be found using the following formula E k = mv 2 2 where m = mass (in kg) and v = speed (in m/s)

Energy changes

Sankey Diagram A Sankey diagram helps to show how much light and heat energy is produced

Sankey Diagram The thickness of each arrow is drawn to scale to show the amount of energy

Sankey Diagram Notice that the total amount of energy before is equal to the total amount of energy after (conservation of energy)

Efficiency Although the total energy out is the same, not all of it is useful.

Efficiency Efficiency is defined as Efficiency (%) = useful energy outputx 100 total energy input

Elastic collisions No loss of kinetic energy (only collisions between subatomic particles)

Inelastic collisions Kinetic energy lost (but momentum stays the same!)

Dog in orbit! The dog is now in orbit! (assuming no air resistance of course)

Dog in orbit! The dog is falling towards the earth, but never gets there!

Dogs in orbit! The force that keeps an object moving in a circle is called the centripetal force (here provided by gravity) Gravity

Other examples Earth ’ s gravitational attraction on moon

Uniform Circular Motion This describes an object going around a circle at constant speed

Direction of centripetal acceleration/force VAVA VBVB VAVA VBVB V A + change in velocity = V B Change in velocity The change in velocity (and thus the acceleration) is directed towards the centre of the circle.

Uniform circular motion The centripetal acceleration/force is always directed towards the centre of the circle Centripetal force/acceleration velocity

How big is the centripetal acceleration? a = v 2= 4π 2 r rT 2 where a is the centripetal acceleration (m.s -2 ), r is the radius of the circle (m), and v is the constant speed (m.s -1 ).

How big is the centripetal force? F = mv 2 r from F = ma (Newton’s 2 nd law) Centripetal Force - The Real Force

Note! There is no such thing as centrifugal force! (at least not until you get to university!) CENTRIFUGAL