Iteration for A Level Maths: staircase method & cobweb method

E.g. f(x) = x² - x - 4 When the above is rearranged in the form of x = g(x), it is said to be in iterative form. x² - x - 4 = 0 x = 1 + x/4 x n+1 = 1 + x n / 4 If x n is known, x n+1 can be calculated. The initial value to trigger off the iterative process is found by looking for change in sign of…

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Simple Harmonic Motion for A Level Physics: vertical motion of a spring

Upward force, T', when the stretched spring is released = k(e+x) If the resultant force is F, F = K(e+x) -mg F = ma - Newton's Second Law k(e+x) - mg = ma, where a is the acceleration (ke + kx) - mg = ma Since T = ke mg - mg + kx = ma ma = kx a = k/m x a ∝ x ω 2  = k/m T =2π √(m/k) T =2π √(m/k) Since the a…

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Mechanics - A Level Applied Maths & Physics: really challenging questions

An object that falls off a tree travels 16/25 th of the height of the tree during the last second. Assuming that g = 10 m/s², find the total time taken for the fall and the height of the tree. A balloon is steadily rising at 5 m/s. After 4 seconds of its motion, a nail from the balloon falls off. Sk…

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Simple Harmonic Motion for A Level Physics: proof of motion of simple pendulum being SHM

As you can see, when the weight of the pendulum bob is resolved, the tension of the string,  T , and the  mg cos x  cancel each other out, leaving  mg sin x  as the net force, as shown above. This force is responsible for bringing the bob down in a curved path. Using F = ma for the bob, mg sin x = ma, wher…

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Practise straight line graphs for GCSE, IGCSE and GCE - OL: fully interactive

This applet generates a straight line at the click of a button. From the line on a grid, you can calculate both the gradient and intercept, if the latter is visible. Otherwise, you need to find the gradient of the line and the coordinates of any point through which the line passes. E.g. If the gradie…

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Basic Integration: finding the area under a curve by a GeoGebra applet - fully interactive

With the following GeoGebra applet, you can find out the area under a curve by integration -  interactively.  All you need to do is  to move the two sliders to points of your choice and tick the checkbox for answer to appear. The function used in this case is a quadratic function. Enjoy, learn and pro…

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Linear and Geometric Sequences for GCSE, IGCSE & A Level

E.g.1 : Find the nth term of the sequence: 3, 7, 11, 15, ... The common difference is 4.  Let N = an + b, where N is the nth term and a an b are constants to be found. 3 = a(1) + b = a + b 7 =  a(2) + b = 2a + b Solving the two equations simultaneously, we get a = 4 and b = -1 So, the nth term, N = 4n - 1 …

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