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• Acceleration, Projectiles & Kinematics

Velocity is the rate of change of an object's position, and therefore has direction. This makes it a vector quantity, unlike speed which is scalar (magnitude but no direction). velocity = distance / time The units of velocity are m/s Acceleration is the rate of change of velocity, and so is the mathematical derivative of this. acceleration = change in velocity / change in time The units of acceleration are m/s² Motion Graphs We can plot motion on two main types of graph - it is important to know the properties of each. Displacement-Time Graphs The Gradient is the velocity - draw a tangent to find the instantaneous velocity Horizontal line represents zero velocity Velocity-Time Graphs The Gradient is the acceleration The Area beneath the graph is the displacement Constant Acceleration When acceleration is constant (e.g. free fall when we ignore air resistance), we can use SUVAT equations to work out the variables: s is for displacement u is for initial velocity v is for final velocity a is for acceleration t is for time Vertical Motion due to Gravity The gravitational force of the earth causes all objects to accelerate towards the ground, its surface. Ignoring air resistance, the acceleration is constant, and given a g: g = 9.8 m/s² This is independent of the mass, shape or velocity of the object. It is vital to set a positive direction of motion for each question In the example above, we set up as the positive direction, so our values for a and v were negative because the ball is going down. Constant Acceleration with Vectors Additionally, we can express motion using vectors: r = r₀ + v t r is the position vector of the moving object r₀ is the initial position vector v is the velocity vector Four of the five SUVAT equations have vector equivalents: v = u + at v = u + a t s = ut + ½at² s = u t + ½ a t² + r₀ s = vt - ½at² s = v t - ½ a t² + r₀ s = ½(u+v)t s = ½( u + v )t + r₀ v² = u² + 2as has no vector equivalent Projectile Motion When we model a projectile, we ignore air resistance. This means that: Horizontal motion of a projectile has constant velocity: a = 0 Vertical motion of a projectile motion is modelled as gravitational free fall: a = g For horizontal projection, like in the diagram above: Since horizontal velocity is constant, we can use the equation x = vt (were x is horizontal displacement) For vertical velocity, we need to use SUVAT equations, due to the constant acceleration of g = 9.8 m/s² Horizontal and Vertical Components When given the velocity as a vector or at an angle, you must use trigonometry to find the vertical and horizontal components. Then, treat them separately as above (the horizontal component still has constant velocity) For an object projected at velocity U at an angle of θ to the x-axis: the horizontal component of the velocity is given as U cos(θ) the vertical component of the velocity is given as U sin(θ) The vertical component is decelerating at -9.8 m/s² as it rises, and accelerates at 9.8 m/s² as it falls The projectile reaches its maximum height when the vertical component of the velocity is zero. Applying SUVAT to the vertical component of the projectile gives us a few standard and useful equations: Projectile motion can also be plotted and calculated with in vectors. Variable Acceleration As you can see from the velocity-time graph, varying acceleration produces curves: This is because the gradient of a velocity-time graph is the acceleration, and so if the acceleration changes with time, so must the gradient. Kinematics Velocity is the rate of change of displacement. Acceleration is the rate of change of velocity. Therefore: This means we can differentiate and integrate equations for motion: Integrating and Differentiating Vectors Variable acceleration can also be expressed in vector form. Therefore, you need to be able to differentiate and integrate vector equations. Often, dot notation is used to quickly represent differentiation with respect to time: To integrate vectors: In both differentiation and integration of vectors, you must do one term at a time See Notes Sheets on differentiation and integration in Pure Maths if you need a recap.

• Modelling in Mechanics

The real world of mechanics is incredibly complicated, with hundreds of different factors affecting motion and stability. This would be near impossible to calculate at this stage, so to simplify it we model objects and scenarios in a number of different ways. Modelling Assumptions Particle Has negligible dimensions Mass acts about a single point Rotational forces and air resistance can be ignored Rod The diameter is negligible (so no thickness), only length counts Mass acts along a line It is rigid, so does not bend or buckle Lamina An object with only area, thickness is negligible (like a sheet of paper) Mass acts across the flat surface Uniform Body The Mass is distributed evenly throughout the object The mass is modelled to act through one point, the centre of mass, in the geometrical middle of the body Light Object The mass of the object is negligible Often used for strings or pulleys Allows us to model the tension on each end of a string as equal Inextensible String A string that does not stretch when a force is applied Allows us to model the acceleration of two connected particles as the same Smooth Surface There is no friction between the surface and any object on it Rough Surface There is friction between the surface and any object on it Wire A rigid length of metal with negligible thickness (only modelled with length) Smooth and Light Pulley The pulley has no mass No friction in the pulley, so tension is the same either side of it Bead A particle with a hole in it for string/wire to be threaded through It can move freely along the wire or string Peg An object from which a body can be hung or rested Modelled as dimensionless and fixed Can be either rough or smooth Air Resistance Resistance forces due to motion through air Almost always modelled as negligible Gravity The force of attraction between all objects Acceleration due to gravity is given by g, which is 9.8 m/s² unless otherwise stated Assume objects are always attracted to the ground (earth) Gravity acts uniformly and downwards Quantities & Units Quantities can be either scalar or vector. Scalar quantities have only magnitude Vector quantities have both magnitude and direction SI Base Units Quantity Unit Symbol Mass kilogram kg Distance metre m Time seconds s Distance is scalar - the vector version of it is displacement Derived Units Quantity Unit Symbol Velocity metres per second m/s Acceleration metres per second² m/s² Force Newtons N Velocity is the vector form of speed, as velocity has direction as well as magnitude Weight is not the same as mass. Weight is a force, measured in Newtons, N. It is given by: W = mg Weight = mass x gravity Representing Forces Weight acts down from an object's centre of gravity. Other forces can act in any direction, and so to avoid confusion it is best to draw a force diagram: The length of arrow should represent the magnitude of the force. Single headed arrows represent forces. If a system is in equilibrium - meaning there is no net resultant force and it is motionless or at constant velocity - then all the force arrows will make a closed shape. The example above would give a triangle.

• Forces, Friction & Motion

As seen in the Notes Sheet on Modelling in Mechanics, forces on an object can be expressed with a force diagram: Newton's 1st Law: On this diagram, you can see that T, the tension force pulling the object to the right, is equal to F, the friction between the the object and the surface - friction is the force that opposes motion. This means that there is no resultant force to the left nor right, so the object remains motionless (it is in equilibrium). This is Newton's first law: An object will stay at a constant velocity, or motionless, unless it experiences a resultant force. Newton's 2nd Law: Here, the surface is smooth (no friction). This means the object experiences a resultant force, T, to the right. According to Newton's second law, the resultant force on an object is directly proportional to the rate of change of momentum of the object, and acts in the same direction. This is commonly expressed as: F = ma Resultant Force = Mass x Acceleration (in direction of resultant force) Newton's 3rd Law: In this example, the object sits at rest on the surface with no horizontal forces acting on it. However, its weight, W, (mass x g) is a force acting down. If this were the only force, you would expect the object to accelerate down through the surface but it does not. This means there is an equal but opposite force acting upwards: the normal reaction, R. For every action, there is an equal but opposite reaction (equal in magnitude and type of force). Forces in Two Dimensions Often, forces will not be exactly horizontal or vertical. Instead, they may be given as vectors or at angles. Sometimes you will need to find the resultant from two forces at right angles to one another, but more often you will be given the resultant and need to calculate the components. Use Pythagoras to do this. When given the resultant at an angle, use trigonometry to find the components. Moments Moments are the turning forces acting on an object, around a pivot. M = Fx moment = force × perpendicular distance from pivot The units of moments are Nm If there is a resultant moment in either direction (clockwise or anti-clockwise), then the object will rotate in that direction about its pivot. When an object is in equilibrium, there is neither a net force nor a net moment in any direction. The Principle of moments can be used for this: the sum of all clockwise moments must equal the sum of all anticlockwise moments for an object to be in equilibrium. When we model a rod or object as uniform, it means we assume all the weight is evenly distributed throughout, and only acts about the centre. Tilting The point of tilting is the point at which a rigid body is about to pivot. When a rigid body is at the titling point, the reaction at any other support is zero. Friction Friction is the force that opposes motion. No materials are perfectly smooth, but have a degree of roughness: this is defined by the coefficient of friction, μ, which is a fixed property of a surface in certain conditions (conditions matter, because if a surface is wet, its coefficient of friction will be lower). An object will only move on a surface if the force applied to it is greater than the maximum friction, F(max): this value is knows as the limiting value. The limiting value, F(max), depends on both the normal reaction force, R, and the coefficient of friction, μ: F(max) = μR Limiting Value = maximum friction = μR Static Rigid Bodies A rigid body is something like a rod or a ladder, leaning against a wall. If it is static, it is in equilibrium. For a rigid body to be in equilibrium: It must be stationary There must be no resultant force in any direction There must be no resultant moment A common example is a ladder leaning against a wall, where there is friction on the ground: Connected Particles & Pulleys When the particles are moving along the same straight line, they can be modelled as one particle with a combined mass. When the particles are moving separately, for example in a pulley system, they can be treated as separate particles. In both instances, when the string is light and inextensible, the acceleration in each particle will be the same. A common question will consist of two connected particles, one on an inclined plane, the other hanging off a pulley:

• Data

• Correlation & Regression

Bivariate data is data with two variables, and can be represented in a scatter diagram. We can describe the correlation between the two variables based on how much of a straight line the points on the diagram form. Correlation describes the nature of the linear relationship between two variables. A negative correlation occurs when one variable increases as the other decreases. A positive correlation occurs when both variables increase together. Causation The relationship can be described as causal if a change in one variable induces a change in the other. It is vital to remember that just because there may be a correlation, no matter how strong, between two variables, it does not mean the relationship is causal. Correlation does not imply causation You need to consider the context of the variables and use common sense to decide whether or not there is causation as well as correlation. Measuring Correlation The product moment coefficient, r, is a measure of strength for linear correlation between two variables. It takes values from -1 to 1, where If r = 1 the correlation is perfect and positive If r = 0 there is no correlation at all If r = -1 the correlation is perfect and negative You calculate the product moment coefficient using a stats-equipped scientific calculator. On a CASIO ClassWiz fx-991EX, to calculate the product moment coefficient, r: Click MENU Click 6: statistics Click 2: y=a+bx Input your data in the table Click AC Click OPTN Click 3: Regression Calc r is the product moment coefficient Linear Regression The line of best fit on a scatter diagram approximates the relationship between the variables. The most accurate form of line of best fit is the least squares regression line, which minimises the sum of the squares of the distances from each data point to the line. The regression line is plotted in the form y = a + bx Where b tells you the change in y for each unit change in x. If the correlation is positive, so is b, and vice versa. To calculate a and b, use your calculator and follow the steps above for the product moment coefficient. Independent & Dependent Variables The independent variable is the one that is being changed, the dependent variable is the one being measured and recorded. The independent variable should always be plotted on the x-axis The dependent variable should always be plotted on the y-axis You should only ever use the regression line to make predictions for the dependent variable Exponential Models Exponentials and logarithms can be used to model non-linear data that still has a clear pattern. If the equation is in the form y = axⁿ, a graph of log(y) against log(x) will give a straight line where log(a) is the y intercept and n the gradient. If the equation is in the form y = ab^x, a graph of log(y) against x will give a straight line where log a is the y intercept and log b the gradient.

• Probability

Probability is used to predict the likeliness of something happening. It is always given between 0 and 1. An experiment is a repeatable process that can have a number of outcomes An event is one single or multiple outcomes A sample space is the set of all possible outcomes For two events, E₁ and E₂, with probabilities P₁ and P₂ respectively: To find the probability of either E₁ or E₂ happening, add the two probabilities, P = P₁ + P₂ To find the probability of both E₁ and E₂ happening, multiply the two probabilities, P = P₁ x P₂ The sample space for rolling two fair six-sided dice and adding up the numbers that show would look like this: To work out the probability of getting a particular result, you count how many times the result occurs and divide by the total number, 36 (since 6² = 36). So to work out the probability of getting a 10, count the number of tens and divide by 36: 3/36 = 0.0833 Generally, give your answers as decimals to three significant figures Conditional Probability If the probability of an event is dependent on the outcome of the previous event, it is called conditional. Conditional probability is noted using a vertical line between the events: The probability of B occurring, given that A has already occurred is given by P(B|A) For two independent events: P(A|B) = P(A|B') = P(A) Experiments with conditional probability can be calculated using a two-way table/restricted sample space: Venn Diagrams A Venn diagram is used to represent events happening. The rectangle represents the sample space, and the subsets within it represent certain events. Set notation is used to describe events within a sample space: A ∩ B represents the intersection A ∪ B represents the union Adding a dash, ', means the compliment, or "not" The addition formula is very useful: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) Mutually Exclusive Events As you can see from the Venn diagrams, mutually exclusive events do not intersect. This means there is no overlap, so: P(A or B) = P(A) + P(B) Independent Events When one event has no effect on the other, the two events are described as independent. Therefore: P(A ∪ B) = P(A) x P(B) Conditional Probability in Venn Diagrams You can find conditional probability easily from Venn diagrams using the multiplication formula: P(B|A) = P(B ∩ A) / P(A) Tree Diagrams Tree diagrams are used to show the outcomes of two or more events happening, one after the other. For example, if there are 3 red tokens and 7 blue tokens in a bag, and two are chosen one after the other without replacement (the first is not put back into the bag), a tree diagram can model this: When you have worked out the probability of each branch, add them together - if they sum to 1, it is correct. Conditional Probability in tree diagrams Tree diagrams show conditional probability in their second and third etc columns. The multiplication formula still applies: P(B|A) = P(B ∩ A) / P(A)

• Statistical Distributions

• Hypothesis Testing

A hypothesis is a statement that has yet to be proved. In statistics, the hypothesis is about the value of a population parameter, and can be tested by carrying out an experiment or taking a sample of the population. The test statistic is the result of the experiment / the statistic generated from the sample. In order to perform a hypothesis test, two hypotheses are required: The null hypothesis, H₀ is the one you assume to be correct The alternative hypothesis, H₁ is the one you are testing for, to see if the assumed parameter is correct or not. A specific threshold for the probability of the test statistic must also be defined. If the probability of the test statistic is lower than this threshold, there is sufficient evidence to reject H₀. If it is above the threshold, there is insufficient evidence to reject H₀. This threshold is known as the significance level, and is typically set at 1, 5 or 10%. When ending a hypothesis test, you must conclude by saying whether or not there is sufficient evidence to reject H₀. Do not say accept or reject H₁ Critical Regions & Values If the test statistic falls within the critical region, there is sufficient evidence to reject H₀. The critical value is the first value to fall inside the critical region. The acceptance region is the set of values that are not in the critical region, so there is insufficient evidence to reject H₀. The actual significance level is the probability of incorrectly rejecting the null hypothesis. What this actually means is that: the actual significance level is the probability of getting the critical value One- and Two-Tailed Tests Hypothesis tests can be one-tailed or two-tailed. This refers to how many critical regions there are: For a one-tailed test, H₁: p < ... or H₁: p > ... and there is only one critical region For a two-tailed test, H₁: p ≠ ... so there are two critical regions, one on each 'tail' See the examples below. Hypothesis Tests on Binomial Distributions Often, hypothesis tests are carried out on discrete random variables that are modelled with a binomial distribution. One-Tailed Example A discrete random variable, X, is distributed as B(12, p). Officially, X is distributed with a probability of 0.45. However, there is a suspicion that the probability is, in fact, higher. Find, at the 5% significance level, the critical region and actual significance level of the hypothesis test that should be carried out. Write out the hypotheses & test statistic H₀: p = 0.45 H₁: p > 0.45 X∼B(12, p) Since we are only looking at whether or not the probability is more than 0.45, it is a one-tailed test. Therefore, look for the first value of X for which the cumulative probability is more than 0.95 (1 - 0.05, the 5% significance level) As you can see, the first value to have a cumulative probability of more than 0.95 is 8, so: The critical value is 8 The critical region is > 7 Find the actual significance level 1 - 0.964 = 0.036 0.036 = 3.6 % The actual significance level is 3.6% Write a conclusion If the experiment were repeated 12 times, and 8 or more of the 12 trials were successful, there would be sufficient evidence to reject H₀, suggesting the probability is indeed higher than 0.45 Two-Tailed Example a. A manufacturer of kebab-makers (a kebab-maker-maker, if you will) claims that just 25% of the kebab-makers he makes make low quality kebabs. At the 10% significance level, find the critical region for a test of whether or not the kebab-maker-maker's claim is true for a sample of 12 kebab-makers. Write out the hypotheses and test statistic H₀: p = 0.25 H₁: p ≠ 0.25 X∼B(12, p) We do not know if the probability could be more or less than 0.25, so the test is two tailed. Therefore, divide the significance level by two, and find the critical region. This will be any cumulative probability that is less than 0.05 or more than 0.95 Here you can see the critical region is in two parts, one at each 'tail' of the values; The critical region is X < 1, X > 5 b. A random sample of 12 kebab-makers is taken, and 5 are found to make low quality kebabs. Does this imply the kebab-maker-maker is lying? Method 1 See if 5 is in the critical region 5 is not > 5 not < 1 Conclude 5 does not lie within the critical region for this test (X < 1, X > 5), so there is insufficient evidence to reject H₀ - this implies the kebab-maker-maker is not lying. Method 2 Find the cumulative binomial probability when X=5 When X∼B(12, 0.25), P(X=5) = 0.946 Conclude P(X=5) = 0.945, which is not within the significance level for the test. Therefore, there is insufficient evidence to reject H₀ - this implies the kebab-maker-maker is not lying. Hypothesis Tests on Normal Distributions You can carry out hypothesis tests on the mean of a normally distributed random variable by looking at the mean of a random sample taken from the overall population. To find the critical region or critical value, you need to standardise the test statistic: Then, you can use the percentage points table to determine critical regions and values, or you can use the inverse normal distribution function on a scientific calculator. Example The kebabs that the kebab-maker makes have diameter D, where D is normally distributed with a mean of 4.80 cm. The kebab-maker is cleaned, and afterwards a 50 kebabs are made and measured, to see if D has changed as a result of the cleaning. D is still normally distributed with standard deviation 0.250 cm. Find, at the 5% significance level, the critical region for the test. Write out your hypotheses H₀: μ = 4.8 H₁: μ ≠ 4.8 Assume H₀ is true: Sample mean of D, Ď ∼ N(4.8, 0.25²/50 ) Code data: Z = (Ď - 0.48) / (0.25/√50) Z ∼ N(0, 1) The test is two tailed, so area on each side should be 0.025 (half of 5%): Decode, using ±1.96 (Ď - 0.48) / (0.25/√50) = -1.96 Ď - 0.48 = -0.0693 Ď = 0.411 (Ď - 0.48) / (0.25/√50) = 1.96 Ď - 0.48 = 0.0693 Ď = 0.549 Conclude The critical region is when the sample mean is smaller than 0.411 or larger than 0.549 Hypothesis Tests for Zero Correlation You can determine whether or not the product moment coefficient, p, of a sample indicates whether or not there is likely to be a linear relationship for the wider population using a hypothesis test. Use a one-tailed test if you want to test if the population p is either > 0 or < 0 Use a two tailed test if you want to see that there is any sort of relationship, so p ≠ 0 The critical region can be determined using a product moment coefficient table. It depends on significance level and sample size. To calculate the product moment coefficient of the sample, use your calculator (see notes sheet on regression & correlation).

• Indices & Algebraic Methods

• Quadratic & Simultaneous Equations and Inequalities