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This post will be a review of the physics topic hoping to aid you in the test tomorrow.

Newton's three laws of motion:

1. Law of Inertia: An object will stay at rest or move in a straight line with a constant velocity unless acted upon by an external force.
2. F=ma An object of a given mass will accelerate at a rate proportionate to the force.
3. For every action, there is an equal opposite reaction.

Speed vs. Velocity

• Speed is scalar (size only)
• Speed = Distance/Time
• Velocity is a vector (size and direction)
• Velocity = Displacement/Time

Displacement is the shortest distance between the starting point and the current point while distance is the total path traveled.

Acceleration is the rate the velocity changes with respect to time.
Formula for average acceleration is (difference in velocity)/time.

Equations of motion:

Where v is the final velocity, u is the initial velocity, a is acceleration and t is time

Where r is displacement.

Newton's Universal Gravitational Law
The force between two bodies can be calculated by using the following formula

Where m1 and m2 are the masses of the two bodies and r is the distance between the centre of the two bodies. G is the Universal gravitational constant.

The gravity of a body can be calculated by using the following formula

Centripetal Force
In centripetal motion, the acceleration is equal to the velocity squared divided by the radius.
Therefore F=ma become F=m(v^2/r)

Calculating Average Acceleration using a ticker timer.
A ticker timer will make 50 marks per second in other words, the time between the dots is equivalent to 1/50 of a second. To find the average acceleration you would have to find the initial velocity (distance between the first two dots/(1/50)) and the final velocity (distance between last to dots/(1/50)). Find the difference between the two velocities and then divide that by the total time taken.

Other notes:
The pronumeral for distance is 's' not 'd'
Displacement is 'r' not 'd'
Speed/Velocity is 'v'not 's'

Good luck for the test tomorrow

Simultaneous Equations

I am writing this post because I've seen several problems with the solving of simultaneous equations.
Eg. If xy = 6 and x+y = 5 find x and y
      x = 5-y
      5y-y^2 = 6
      (y-2)(y-3)=0
      y= 2 or 3
If y = 2, x=3 and if y = 3, x =2

If you are trying to solve a simultaneous equation with a function and a non-function, then final substitution must be made into the function otherwise it may yield incorrect results.
Eg. Solve simultaneously
      x^2 + y^2 = 16 (circle graph - not a function)
      3x - 4y - 20 = 0
      y = (3x-20)/4
Substituting into the first equation gives you
x^2 + [(3x-20)/4]^2 = 16
x^2 + (9x^2 - 120x + 400)/16 = 16
16x^2 + 9x^2 - 120x + 400 = 256
25x^2 - 120x + 144 = 0
(5x - 12)^2 = 0
x = 12/5 At this point, x must be substituted into the function to find the y value, otherwise it will give more than one result because there are two y values for just about every x value.

Another note sqrt(x^2 + y^2) ≠ x+y
(x + y)^2 ≠ x^2 + y^2

Introductory Calculus

Calculus is a major part of mathematics that studies change. There are two major branches, namely Differential Calculus and Integral Calculus which are linked together by the Fundamental Theorem of Calculus.

Differential Calculus
Differential Calculus is the study of the rates at which a function changes. In a straight line graph, this would be the gradient, but in curves, it would be the gradient of the tangent at particular points. We find the gradient of these tangents through a process called differentiation. The derivative of a function (function after differentiation) can be written as

There are a few rules for differentiating:

1. Power Rule
   
   
   
   
   
   
   
   
   
   
   
   
2. Product Rule: The y is a product of two functions, then the derivative is:
   
   
   
   
   
   
   
   
3. Quotient Rule: if y is a function divided by another function, then the derivative of y is:
   
   
   
   
4. Chain Rule/Function of a Function:
   
   
   
   
   
   
   
   
   
   
   
   
   
5. Differentiation by first principles: The original but long method
   

Integral Calculus
Integral Calculus is the study of areas under functions. In a straight line graph, this would be a trapezium, but in curves it can be very hard to find the exact value, integral calculus can give the exact value. There are two different types of Integrals (Function after integrating). They are the Indefinite (also the antiderivative, opposite of derivative) and Definite (which give the area between the curve and the x axis within the boundaries).

The integral of a function is:

Hope you have learnt something!

Equations and Inequalities

Linear Equations
Linear equations are just equations with a degree of one (highest power of a variable is one). They can be solved simply by rearranging the terms.
Eg. 7 + 5x = -8
      5x = -15
      x = -3

      3(9 + x) = 6
      9 + x = 2
      x = -7

Equations with Squares and Roots
Equations with squares or roots can be solved like a normal equation. In some case, you may have to use methods to solving quadratic equations.

1. Completing the square
   Adding the square of half of the coefficient of x
   Eg. x^2 -4x -12 = 0
   x^2 -4x + 4 = 16
   (x - 2)^2 = 16
   x - 2 = +/-4
   x = 6 or -2
2. Factorising
   x^2 -4x - 12 = 0
   (x-6)(x+2)=0
   x = 6 or -2
3. Quadratic Formula
   

Inequalities
Inequations can be solved just like an equations, but when ever you divide or multiply by a negative number, the sign will be flipped around.
Eg. -(x+2) > -5
      x+2 < 5
      x < 3

Simultaneous Equations
There are two methods to solve simultaneous equations but I will recommend you use the substitution method.
Eg. x^2 + 2x = 5x - 2
      x^2 - 3x + 2 = 0
      (x-2)(x-1) = 0
      x = 1 or 2

Inequalities with an unknown denominator
A bit more advanced stuff. If you had a unknown in the denominator you would have to find critical points in the inequation. A critical point is a point where the gradient of the tangent is zero or cannot be defined. The denominator of a fraction cannot be zero in this case when the denominator is equal to zero gives the critical point. The inequation can then be solved afterwards and then substitution of points is necessary. (I think I confused you)
Eg.

Find the Critical Points

Solve for x

Substitute values in to determine whether it fits in.

Hope this has helped you.

Rules of Circle Geometry

The circle is the most fascinating shape in mathematics. The circle is a locus of all the points that are the same distance from one point.

There are twelve rules in circle geometry

1. Equal arc/chord subtend equal angles at the centre. Equal angles stand on an equal arc/chord.
   
2. Equal chords are equal distance from the centre. Chords that are equal distance from the centre are equal.
   
3. The line from the centre of a circle to the centre of a chord is perpendicular to the chord. A perpendicular line from the chord to the centre bisects the chord.
   
4. The angle at the centre is twice the angle standing on the same chord/arc.
   
5. Angles on the same arc are equal.
   
6. Angles on a semi-circle is 90°
   
7. Cyclic Quadrilaterals are quadrilaterals where all four corners lie on a circle. The opposite angles of cyclic quadrilaterals are supplementary.
   
   x+y=180°
8. The angle between the radius and a tangent is 90°
   
9. Tangents drawn from an external point are equal.
   
10. Two chords or secants (AB and CD) intersecting at X, then AX.BX = CX.DX
    
11. A secant and a tangent (ABX and CX) intersecting at X, then AX.BX=CX^2
    
    
    
12. The angle between a chord and a tangent is equal to the angle in the alternate segment.
    

Hope you have learnt something.