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Motion and Energy

M H Inglés

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Motion and Energy

Speed and Velocity
- Speed is just how fast you're going (e.g. 30 mph or 20 m/s) nothing to do with direction
- Velocity is speed with the direction specified (e.g. 30 mph north or 20 m/s, 060°)
Speed = distance ÷ time
- The gradient of a distance-time graph shows how fast an object is travelling
  - The gradient is the change in the distance divided by the change in time
- Speed = gradient
  - Vertical -------------- Horizontal
    - 500 ------ 30
      - = 16.7 m/s
- Straight uphill or straight downhill sections mean the object is travelling at a steady speed
  - The steeper the graph the faster the object is going
- Downhill sections mean the object is going back toward its starting point
- Curves represent acceleration or deceleration
  - A steepening curve means the object is speeding up (increasing gradient)
  - A levelling off curve means the object is slowing down (decreasing gradient)
- Flat sections are when the object is stationary
Acceleration
- Acceleration is how quickly the velocity is changing. This change in velocity can be a change in speed or a change in direction or both
- Acceleration = Change in Velocity ----------------------------- Time taken
  - (v-u) ------ a|t
    - 'v' is the final velocity, 'u' is the initial velocity, 'a' is acceleration and 't' is time taken
- Acceleration is the change in velocity (m/s) per second (s) = m/s²
- Velocity-Time graphs
  - Calculating acceleration, velocity and distance from a velocity-time graph
    - Acceleration = gradient = vertical change ÷ horizontal change
    - The velocity at any point is simply found by reading the value off the velocity axis
    - The distance traveled in any time interval is equal to the are under the graph - e.g. the distance travelled between 20s and 40s is equal to the area of 20 x 20
      - 400m
  - Gradient = Acceleration
  - Flat sections represent steady speed
  - Uphill sections are acceleration
  - Downhill sections are deceleration
  - The area under any section of the graph is equal to the distance travelled in that time
  - A curve means changing acceleration
Gravity
- Gravity attracts all masses but mostly large things such as planets - anything near a star or planet is attracted to it very strongly
  - This means that on the surface of a planet it makes all things accelerate towards the ground and it gives everything a weight
- Weight and mass are not the same
  - Mass is just the amount of 'stuff' in an object
    - For any given object this will have the same value anywhere in the universe
    - Mass is not a force and is measured in kilograms
  - Weight is caused by the pull of the gravitational force
    - Weight is a force measured in newtons
  - Weight = mass x gravitational field strength
    - W= m x g
    - On Earth g ≈ 10 N/kg
    - On the moon g ≈ 1.6 N/kg
Any resultant force will produce acceleration
- F = ma or a = F/m
  - m = mass (kg) a = acceleration (m/s²) F = resultant force (N)
Resultant Forces
- Stationary object = all forces balanced
  - -The force of gravity (weight) is acting downwards -This causes a reaction force from the surface pushing up on the object -Without a reaction force, it would accelerate downwards due to the pull of gravity -The resultant force on the teapot is zero
- Resultant force is the overall force on a point or object
  - In most situations there are at least two forcesacting on an object along any direction
  - The overall effect of these forces will decide the motion of the object - whether it will accelerate, decelerate or stay at a steady speed
    - If the resultant force on a stationary object is zero the object will remain stationary
    - If there is no resultant force on a moving object it will just carry on moving at the same velocity
    - If there is a non-resultant force then the object will accelerate in the direction of the force
      - This 'acceleration' can be starting, stopping, speeding up, slowing down or changing direction
  - If the forces all act along the same line the overall effect is found by just adding or subtracting them
  - Resultant forces change an object's velocity
Reaction Forces
- When two objects interact the forces they exert on each other are equal and opposite
- When object A (mass 55kg) pushes on object B (mass 65), (the 'action force), there is an equal and opposite force from object B (the 'reaction' force) - both objects feel the same sized force, in opposite directions and so accelerate away from each other
  - Object A will be accelerated more than object B as it has a smaller mass
Frictional Force
- If an object has no force propelling it along it will always slow down and stop because of friction
- Resistance or 'Drag' from fluids (air or liquid)
  - Frictional forces from fluids always increase with speed - a car has much more friction to work against when travelling at 70 mph compared to 30 mph, so at 70 mph the engine has to work much harder just to maintain a steady speed
Terminal Velocity
- When falling objects first set off, the force of gravity is much more than the frictional force slowing them down, so they accelerate
- As the speed increases the friction builds up, this gradually reduces the acceleration until eventually the frictional force is equal to the accelerating force and then it won't accelerate any more
- It will have reached its maximum speed or terminal velocity and will fall at a steady speed
- The accelerating force acting on all falling objects is gravity and it would make them fall at the same rate if it wasnt for air resistance, this means that on the moon where there's no air hamsters and feathers dropped simultaneously will hit the ground together, however on earth air resistance causes things to fall at different speeds and the terminal velocity of any object is determined by its drag in comparison to its weight; the frictional force depends on its shape and area
- One example is the human skydiver, without his parachute open he has quite a small surface area and a force of 'W = mg' pulling him down, he reaches the terminal velocity of about 120mph, but with the parachute open there's much more air resistance and still only the same force 'W = mg' pulling him down, this means his terminal velocity comes right down to about 15mph, which is a safe speed to hit the ground at
Stopping Distances
- Thinking distance
  - How fast you are going
  - How aware you are
    - tiredness, drugs, alcohol or carelessness
- Braking distance
  - How fast you are going
  - How good your brakes are
  - How good the tyres are
  - How good the grip is
    - road surface
    - weather conditions
    - tyres
- 30 mph thinking = 9m braking = 14m
- 50 mph thinking = 15m braking = 38m
- 70 mph thinking = 21m braking = 75m
Work and Potential Energy
- When a force moves an object through a distance energy is transferred and work is done
  - Whenever something moves, something else is providing some sort of 'effort' to move it - the thing putting the effort in needs a supply of energy (like fuel or food or electricity etc) it then does 'work' by moving the object - and one way or another it transfers the energy it receives (as fuel) into other forms - whether this energy is transferred 'usefully' (e.g. by lifting a load) or is 'wasted' (e.g. lost as heat through friction) 'work is still done'
    - 'Work done' and 'energy transferred' are the same thing (and both are given in joules)
- WORK DONE = FORCE X DISTANCE
- Gravitational Potential Energy
  - Gravitational Potential Energy = mass x g x height
  - Gravitational potential energy (measured in joules) is the energy that an object has by virtue of its vertical position in a gravitational field, when an object is raised vertically work is done against the force of gravity and the object gains gravitational potential energy
    - On Earth the gravitational field strength (g) is approximately 10N/kg
Kinetic Energy
- Kinetic energy is energy of Movement
  - Kinetic Energy = 1/2 x mass x speed²
    - The amount of kinetic energy of something depends on both its mass and speed, the more it weighs and the faster its going the greater the kinetic energy will be
- Kinetic energy transferred is work done
  - A moving car can have a lot of kinetic energy, to slow a car down this kinetic energy needs to be converted into other types of energy (using the laws of conservation of energy) - to stop a car the kinetic energy (1/2mv²) has to be converted to heat energy as friction between the wheels and the brake pads, causing the temperature of the brakes to increase
    - Kinetic Energy Transferred = Work Done by Brakes
    - 1/2mv² = F x d
      - m = mass of car and passengers (in kg)
      - v = speed of car (in m/s)
      - F = maximum braking force (in N)
      - d = braking distance (in m)
- Falling objects convert Ep (gravitational potential energy) into Ek (kinetic energy)
  - Some of the Ek (kinetic energy) is transferred into heat and sound
Power
- Power is the rate of doing work
  - Power is not the same thing as force or energy, a powerful machine is not necessarily one which can exert a strong force - a powerful machine is one which transfers a lot of energy in a short space of time
    - Power = Work done (or energy transferred) --------------------------------------- Time taken
      - P = E --- t
- Power is measured in Watts (or J/s)
  - 1 watt = 1 joule of energy transferred per second
- Ways of measuring the power output of a person
  - The timed run upstairs
    - In this case the 'energy transferred' is the potential energy you gain (= mgh), hence Power = mgh/t
  - The timed acceleration
    - This time the energy transferred is the kinetic energy you gain (= 1/2mv²), hence Power = 1/2mv²/t
Forces and Elasticity
- Work done to an elastic object is stored as elastic potential energy
  - When you apply a force to an object you may cause it to stretch and change in shape, any object that can go back to its original shape after the force has been removed is an elastic object - work is done to an elastic object to change its shape, the energy is not lost but is stored by the object bas elastic potential energy, the elastic potential energy is then converted to kinetic energy when the force is removed and the object returns to its original shape e.g. when a spring or an elastic band bounces back
- If a spring is supported at the top and then a weight is attached to the bottom, it stretches - the extension (e) of a stretched elastic object is directly proportional to the load or force applied F, the extension is measured in metres and the force is measured in newtons
  - F = k x e
    - k is the spring constant, its value depends on the material that you are stretching and it's measured N/m
- There's a limit to the amount of force you can apply to an object for the extension to keep increasing proportionally - the graph shows force against extension for an elastic object, for small forces, force and extension are proportional, so the first part of the graph shows a straight-line relationship between force and extension - there's a maximum force that the elastic object can take and still extend proportionally
Momentum and Collisions
- Momentum = mass x velocity
  - Momentum (p) is a property of moving objects, the greater the mass of an object and the greater its velocity the more momentum the object has
    - Momentum is a vector quantity - it has size and direction (like velocity, but not speed)
  - Momentum (kg m/s) = mass (kg) x velocity (m/s)
- Momentum before = momentum after
  - In a closed system the total momentum before an event (e.g. a collision) is the same as after the event - this is called conservation of momentum
    - A closed system means that there are no external forces acting
  - Example Collisions: Two skaters approach each other, collide and off together - at what velocity do they move after collision
    - -choose which direction is positive e.g. positive is to the right, total momentum before collision = momentum of A + momentum of B = A (80kg 2m/s to the right), B (60kg 1.5m/s to the left) = {80 x 2} + {60 x (-1.5)} = 70kgm/s
      - Total momentum after collision = momentum of A and B together = 140 x v, so 140v = 70 i.e. v = 0.5 m/s to the right
  - Example Explosions: A gun fires a bullet - at what speed does the gun move backwards
    - Gun weight = 1kg, bullet weight = 0.01kg, bullet speed = 150 m/s to the right
    - Choose which direction is positive e.g. positive is to the right - total momentum before firing = 0kgm/s - total momentum after firing = momentum of bullet + momentum of gun = (0.01 x 150) + (1 x v)
      - The momentum of a system before an explosion is zero, so, due to the conservation of momentum, the total momentum after an explosion is zero too
        So 1.5 + v = 0 i.e. v = -1.5 m/s, so the gun moves backwards at 1.5 m/s
- Forces cause changes in momentum
  - When a force acts on an object, it causes a change in momentum, a larger means a faster change of momentum (and so a greater acceleration), likewise, if someone's momentum changes very quickly (like in a car crash), the forces on the body will be very large, and more likely to cause injury, this is why cars are designed with safety features that slow people down over a longer time when they have a crash - the longer it takes for a change in momentum, the smaller the force
Car design and Safety
- Brakes and other safety aspects of a car are designed to slow down the rate of kinetic energy transfer and hence reduce the change in momentum
  - Examples of ways cars convert kinetic energy safely are: crumple zones, side impact bars, seat belts and air bags
    - The more powerful a car engine is (kilowatt kw) the more energy it transfers from its fuel every second, so the faster its speed