Velocity and Distance-Time Graphs
If something has velocity it has both speed and direction. Speed and velocity are measured in m/s or km/h or mph. They are both about how fast something is going but speed is purely how fast an object is going, it has no relation to direction. Whereas velocity does e.g. 30 mph north.
Distance-time graphs describe something travelling through time and space. Below are things you need to know:
- Gradient = Speed
- Flat sections are where the object is stationary
- Straight uphill or downhill sections mean it is travelling at a steady speed.
- The steeper the graph, the faster it is going
- Downhill sections mean it's going back to its starting point
- Curves represent acceleration or deceleration
- A steepening curve mean it's speeding up
- A levelling off curve means it's slowing down
To calculate the speed of a distance-time graph just look at the gradient. Divide the change in distance (vertical axis) by the change in time (horizontal axis) to give you the gradient, which is the speed. So there you have it.
Acceleration is simply how quickly the velocity is changing. The change in velocity can be a change in speed or direction or even both. Though you only need to worry about a change in speed for the calculations. The formula for acceleration is change in velocity ÷ time taken = acceleration. However, the formula triangle is a bit tricky, this is what it looks like: (v - u) with a x t underneath it. First of all, there's the (v - u) which means working out the change in velocity, as shown in the example that follows this, rather than putting a simple value for velocity or speed in. Secondly the units, for acceleration it is m/s² instead of m/s as it is for velocity. Heres an example:
A skulking cat accelerates from 2m/s to 6m/s in 5.6s. Find its acceleration. Using the formula triangle: a = (v-u) / t = (6-2) / 5.6 = 4 ÷ 5.6 = 0.71 m/s²
Just remember, acceleration is the change in velocity (m/s) per second (s) = m/s².
Firstly, here are the most important things you need to know about velocity time graphs:
- Gradient = Acceleration
- Flat sections represent steady speed.
- The steeper the graph, the greater the acceleration or deceleration.
- Uphill sections / are acceleration
- Downhill sections \ are deceleration
- The area under any section of the graph (or all of it) is equal to the distance travelled in that time interval.
- A curve means the acceleration is changing.
Below is how to calculate the acceleration, velocity and distance from a velocity-time graph:
Acceleration = gradient = vertical change ÷ horizontal change = your answer.
Velocity at any point is simply found by reading the value off the velocity axis.
The distance travelled in any time interval is equal to the area under the graph line.
Gravity Gravitational force is the force of attraction between all masses. Gravity attracts all masses, but you only notice it when one of the masses is huge, e.g. a planet. Anything near a planet or a star is attracted to it very stongly, which has two important effects:
1. On the surface of a planet, it makes all things accelerate towards the ground (all with the same acceleration, g, which is about 10 m/s² on Earth).
2. It gives everything a weight
Weight and Mass
Weight and mass are not the same thing. Mass is just the amount of 'stuff' in an object, for any given object this will have the same value anywhere in the universe. Whereas, weight is caused by the pull of gravitiaional force. In most questions the wieght of an object is just the force of gravity pulling it towards the centre of the Earth, An object will have the same mass whether it is on Earth or on the moon - but it's weight will be different. A 1kg mass will weigh less on the moon (about 1.6N) than it does in Earth (about 10N), simply because the gravitational force pulling on it is less. Weight is a force measured in newtons. It's using a spring balance or newton meter. Mass is not a force. It's measured in kilograms with a mass balance (an old-fashioned pair of balancing scales).
Formula for Gravity, Mass and Weight
weight = mass x gravitational field strength or w = m x g
Since weight and mass are not the same thing they are measured in different units. Mass is in kg and weight is in newtons (N). The letter g represents the strength of the gravity and its value is different for different planets. On Earth g is roughly 10 N/kg. On the Moon, where gravity is weaker, g is only about 1.6 N/kg. Below is an example of how to put the formula into use.
What is the weight, in newtons, of a 5kg mass, both on Earth and on the Moon?
w = m x g On Earth: W = 5 x 10 = 50N (the weight of the 5 kg mass is 50N)
On the Moon: W = 5 x 1.6 = 8N (the weight of the 5 kg mass is 8N)
So, that proves how easy it is. As long as you know that the gravitational field strength on the Moon is 1.6N and 10N on Earth and you remember what all the letters mean, you'll be able to answer the questions.
The resultant force is the overall force on a point or object. In most real situations there will be at least two forces acting upon 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 there are a number of forces acting at a single point, you can replace them with a single force (so long as the single force has the same effect in the motion as the original forces acting all together). If the forces all act along the same line (they're all parallel and act in the same or the opposite direction), the overall effect is found is found by just adding or subtracting them. The overall force you get is called the resultant force. For example, a stationary teapot. The force of gravity (10N) is acting downwards, this causes a reaction force (10N) from the surface pushing up on the object. This is the only way it can balance. Without a reaction force, it would accelerate downwards due to the pull of gravity. The resultant force on the teapot is therefore zero: 10N - 10N = 0N. A resultant force means a change in velocity. If there is a resultant force acting on an object, then the object will change its state of rest or motion. In other words it causes a change in the object's velocity.
Finding the Resultant Force
By using the method in the example below, you will be able to find the resultant force acting in a straight line.
Benny is driving to Las Vagas in his vintage sports car. He applies a driving force of 1000N, but has to overcome air resistance of 600N. What is the resultant force? Will the car's velocity change?
Say that the forces pointing to the left are pointing in the positive direction. The resultant force = 1000N - 600N = 400N to the left. If there is a resultant force then there is always an acceleration, so Benny's velocity will change (See the diagram on page 46 in the CGP physics revision guide for a picture to support this method).
There you have it. So basically, working out resultant forces is just adding and subtracting as long as you've accounted for everything.
Forces and Acceleration
An object needs a force to start moving. If the resultant force on a stationary object is zero, the object will remain stationary.
No resultant force means no change in velocity. If there is no resultant force on a moving object it'll just carry on moving at the same velocity. For example, when a train/bus/car/anything else is moving at a constant velocity then the forces on it must all be balanced. Never think that things need a constant overall force to keep them moving, to keep going at a steady speed there must be zero resultant force. Make sure you know that.
A resultant force means acceleration. If there is a non-zero resultant force, then the object will accelerate in the direction of the force. A non-zero resultant force will always produce acceleration or deceleration which could be up to five different forms: starting, stopping, speeding up, slowing down or changing direction. On a force diagram this will be shown by unequal arrows.
DON'T EVER WRITE ON A PAPER THAT 'IF SOMETHING IS MOVING THERE MUST BE AN OVERALL RESULTANT FORCE ACTING ON IT.' THAT IS INCORRECT. IF THERE'S AN OVERALL FORCE IT WILL ALWAYS ACCELERATE. YOU GET STEADY SPEED WHEN THERE IS ZERO RESULTANT FORCE.
Forces and Acceleration 2
A non-zero resultant force produces an acceleration. The formula for this is F = ma or a = F/m.
m means mass in kilograms (kg)
a means acceleration in metres per second squared (m/s²)
F is the resultant force in newtons (N)
Here's an example:
a car of mass of 1750 kg has an engine which provides a driving force of 5200 N. At 70 mph the drag force acting on the car is 5150 N. Find its acceleration a) when first setting off from rest b) at 70 mph
First draw a force diagram for both cases. Then work out the resultant force and acceleration of the car in each case. a) Resultant force = 5200 N 5200 ÷ 1750 = 3.0 m.s² b) Resultant force = 5200 - 5150 = 50N 50 ÷ 1750 = 0.03 m/s²
Reaction force are equal and opposite. When two objects interact, the forces that exert on each other are equal and opposite. Which means that if you push something, for example a shopping trolley, the trolley will push back against you, just as hard. As soon as you stop pushing, so will the trolley. The slighty difficult thing to understand is that if the forces are always equal, how does anything ever go anywhere? The important thing to remember is that the two forces are acting on different objects. Think about a pair of ice skaters:
When skater A pushes on skater B (the action force) she feels an equal and opposite force from skater B's hand (the reaction force). Both skaters feel the same sized force, in opposite directions, and so accelerate away from each other. The skater with a smaller mass will be accelerated more because of the formula a = F/m.
It's similar to going swimming. You push back against the water with your arms and legs, and the water pushes you forwards with an equal-sized force in the opposite direction.
Friction is always there to slow things down. If an object has no force propelling it along it will always slow down and stop because of friction (unless you’re in space where there’s nothing to rub against). Friction always acts in the opposite direction to movement. To travel at a steady speed, the driving force needs to balance the frictional forces. You get friction between two surfaces in contact, or when an object passes through a fluid (drag). Most of the resistive forces are caused by air resistance or “drag”. The most important factor by far in reducing drag in fluids is keeping the shape of the object streamlined. The opposite extreme is a parachute which is about as high drag as you can get - which is, of course, the whole idea. Drag increases as the speed increases. Frictional forces from fluids always increase with speed. A car has much more friction to work against when travelling at 70mph compared to 30mph. So at 70mph the engine has to work much harder just to maintain a steady speed.
Objects falling through fluids reach a terminal velocity. When falling objects first set off, the force of gravity is much more than 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 terminal velocity of falling objects depends on their shape and area. The accelerating force acting in all falling objects is gravity and it would make them all fall at the same rate, if it wasn’t 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. The most important example is the human skydiver. Without his parachute open he has quite a small area and a force of ‘W = mg’ pulling him down. He reaches a terminal velocity of about 120mph. But with his parachute open, there is much more air resistance (at any given speed) 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.
Many factors affect your total stopping distance. Looking at things simply - if you need to stop in a given distance, then the faster a vehicle is going, the bigger braking force it’ll need. Likewise, for any given braking force, the faster you’re going, the greater your stopping distance. But in real life it’s not quite that simple - if your maximum braking force isn’t enough, you’ll go further before you stop. The total stopping distance of a vehicle is the distance covered in the time between the driver first spotting a hazard and the vehicle coming to a complete stop. The stopping distance is the sum of the thinking distance and the braking distance.
- The reaction time is the time between a driver spotting a hazard and taking action.
The distance the vehicle travels during the driver’s reaction time. It’s affected by two main factors:
1. How fast you’re going - obviously. Whatever your reaction time, the faster you’re going, the further you’ll go.
2. How dopey you are - this is affected by tiredness, drugs, alcohol and a careless attitude.
Bad visibility and distractions can also be a major factor in accident - lashing rain, messing about with the radio, bright oncoming lights, etc. might mean that a driver doesn’t notice a hazard until they’re quite close to it. It doesn’t affect your thinking distance, but you start thinking about stopping nearer to the hazard, and so you’re more likely to crash.
The distance the car travels under the breaking distance. It is affected by four main factors:
1. How fast you’re going - the faster you’re going, the further it takes to stop.
2. How good your brakes are - all brakes must be checked and maintained regularly. Worn or faulty brakes will let you down catastrophically just when you need them the most, i.e. in an emergency.
3. How good the tyres are - tyres should have a minimum tread depth of 1.6mm in order to be able to get rid of the water in wet conditions. Leaves, diesel spills and muck on the road can gently increase breaking distance and cause the car to skid too.
4. How good the grip is - this depends on three things: road surface, weather conditions and the tyres.
Wet or icy roads are always much more slippy than dry roads, but often you only discover this when you try to brake hard. You don’t have as much grip, so you travel further before slipping.
Work and Potential Energy
When a force moves an object through a distance, energy is transferred and work is done. This sounds far more complicated than it actually is. So instead think of it like this: 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), you can still say that work is done. So, work done and energy transferred are the same thing. They’re both given in joules too.
Work Done = Force x Distance
Whether the force is friction or weight or tension in a rope, it’s always the same. To find how much energy has been transferred (in joules), you just multiply the force in N by the distance moved in m. It’s that easy, here’s an example to prove it:
Some hooligan kids drag an old tractor tyre 5m over rough ground. They pull with a total force of 340N. Find the energy transferred.
W = F x d = 340 x 5 = 1700 J.
Gravitational Potentail Energy
Gravitational potential energy is energy due to height. Gravitational potential energy (measured in joules) is the energy that an object has by virtue of (because of) its vertical position in a gravitational field. When an object is raised vertically, work is done against the force of gravity (it takes effort to lift it up) and the object gains gravitational potential energy. On Earth the gravitational field strength is approximately 10 N/kg.
Gravitational Potential Energy = mass x g x height
A sheep of mass 47 kg is slowly raised through 6.3m. Find the gain in potential energy.
Just put the numbers into the formula: 47 x 10 x 6.3 = 2961 J
Remember that by lifting something up you do work by transferring chemical energy into gravitational potential energy.
Kinetic energy is the energy of movement. Anything that’s moving has kinetic energy. There’s a slightly tricky formula for it, so you have to concentrate a little bit harder for this one.
Kinetic Energy = ½ x mass x speed ²
A car of mass 2450 kg is travelling at 38 m/s. Calculate its kinetic energy.
½ x 2450 x 38 ² = 1768900 J
Remember, the kinetic energy of something depends both on mass and speed. The more it weighs and the faster it’s going, the bigger its kinetic energy will be. So, small mass, not fast, low kinetic energy and big mass, real fast, high kinetic energy.
Kinetic Energy Transferred
Kinetic energy transferred is work done. When a car is moving it has kinetic energy. It can in fact 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 law of conservation energy). To stop a car, the kinetic energy 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 if the car (kg)
v = speed of car (m/s)
f = maximum braking force (N)
d = braking distance (m)
When something falls, its potential energy is converted into kinetic energy. So the further it falls, the faster it goes. Kinetic Energy gained = Potential Energy lost
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. This energy is not lost but is stored by the object as elastic potential energy. This 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 ban bounces back.
Extension of an elastic object is directly proportional to force. If a spring is supported at the top and then a weight attached to the bottom, it stretches. The extension (e) of a stretched spring (or other 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. K is the spring constant. Its value depends on the material that you are stretching and it's measured in newtons per metre (N/m). But this stops working when the force is great enough. There's a limit to the amount of force you can apply to an object for the extension to keep on increasing proportionally. For small forces, force and extension are proportional. There is a maximum force that the elastic object can take and still extend proportionally. This is known as the limit of proportionality. If you increase the force past the limit, the material will be permanently stretched and when the force is removed the material will be longer than at the start.
Power is the rate of doing work - i.e. how much per second. Power is not the same thing as force, nor energy. A powerful machine is not necassarily on which can exert a strong force (though it usually ends up that way). A powerful machine is one which transfers a lot of energy in a short space of time. This is the very easy formula for power:
power = Work done (or energy transferred) ÷ time taken or P = E ÷ t or E = P x t
Power is measured in Watts or J/s. One watt = 1 joule of energy transferred per second. Power means "how much energy per second" so watts are the same as joules per second. DON'T EVER SAY WATTS PER SECOND THOUGH AS THERE IS NO SUCH THING.
A motor transfers 4.8 kJ of useful energy in 2 minutes. Find its power output.
P = E ÷ t = 4800 ÷ 120 = 40 W
Calculating Power Output
There are a couple of different ways to calculate the power output of a person:
1. The timed run upstairs: In this case the energy transferred is the potential energy you gain (=mgh). Hence Power = mgh ÷ t (see page 54 for further details).
2. The timed acceleration: This time the energy transferred is the kinetic energy you gain (½mv²). Hence Power = ½mv² ÷ t
But to get accurate results from these experiments, you have to do them several times and find an average.
Momentum and Collisions
Momentum (kg m/s) = Mass (kg) x Velocity (m/s)
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 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 the Conservation of Momentum. A closed system is just a fancy way of saying that no external forces act.
Forces cause change in momentum. When a force acts on an object, it causes a change in momentum. A larger force means a faster change in momentum (and so greater acceleration). Likewise, if someones' 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.
So, momentum is always conserved in collisions and explosions when there are no external forces acting.
Car Design and Safety - Brakes
Brakes do work against the kinetic energy of the car. When you apply the brakes to slow down a car, work is done. The brakes reduce the kinetic energy of the car by transferring it into heat (and sound) energy. In traditional braking systems that would be the end of the story, but new regenerative braking systems used in some electric or hybrid cars make use of the energy, instead of converting it all into heat during braking.
1. Regenerative brakes use the system that drives the vehicle to do the majority of the braking.
2. Rather than converting the kinetic energy of the vehicle into heat energy, the brakes put the vehicle's motor into reverse. With the motor running backwards, the wheels are slowed.
3. At the same time, the motor acts as an electric generator, converting kinetic energy into electrical energy that is stored as chemical energy in the vehicle's battery. This is the advantage of regenerative brakes - they store the energy of braking rather than wasting it. It's a nifty chain of energy transfer.
Car Design and Safety - Crashes
Cars are designed to convert kinetic energy safely in a crash. If a car crashes it will slow down very quickly - this means that a lot of kinetic energy is converted into other forms of energy in a short amount of time, which can be dangerous for the people inside. In a crash, there'll be a big change in momentum over a very short time, so the people inside the car experience huge forces that could be fatal. Cars are designed to convert the kinetic energy of the car and its passengers in a way that is safest for the car's occupants. They often do this by increasing the time over which momentum changes happen, which lessens the forces on the passengers.
Crumple Zones at the front and back of the car crumple up on impact. The car's kinetic energy is converted into other forms of energy by the car body as it changes shape. Crumple zones increase the impact time, decreasing the force produced by the change in momentum.
Side Bars Impact are strong metal tubes fitted into car door panels. They help direct the kinetic energy of the crash away from the passengers to other areas of the car, such as the crumple zones.
Seat Belts stretch slightly, increasing the time taken for the wearer to stop, This reduces the forces acting in the chest. some of the kinetic energy of the wearer is absorbed by the seat belt stretching.
Air bags also slow down more gradually and prevent you from hitting hard surfaces inside the car.
Car Design and Safety - Power Ratings
The size and design of car engines determine how powerful they are. The more powerful an engine is, the more energy it transfers from its fuel every second, and so the faster its top speed can be. E.g. the power output of a typical small car will be around 50 kW and a sports car will have be about 100 kW (some are much higher). Cars are also designed to be aerodynamic. This means that they are shaped in such a way that air flows very easily and smoothly past them, so minimising their air resistance. Cars reach their top speed when the resistive force equals the driving force provided by the engine. So, with less air resistance to overcome, the car can reach a higher speed before this happens. Aerodynamic cars therefore have higher top speeds.