Physics

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27 01, 2017

Motion in One Dimension

By |2017-01-27T15:40:06+10:00January 27th, 2017|Tags: , , , |0 Comments

Motion in One Dimension Proof of $$a=v.\frac{dv}{dx}$$ we know $$a=\frac{dv}{dt}$$ and $$v=\frac{dx}{dt}$$ $$a=v.\frac{dv}{dt}.\frac{1}{v}$$ since $$v=\frac{dx}{dt}$$, then $$\frac{1}{v}=\frac{dt}{dx}$$ $$a=v.\frac{dv}{dt}.\frac{1}{v}$$ ⇒ $$a=v.\frac{dv}{dt}.\frac{dt}{dx}$$ ∴ $$a=v.\frac{dv}{dx}$$ worked examples further examples = 10x + 15 - 7x + 14 = 3x + 29 harder examples u = 1 [...]

17 10, 2016

Differential Proof

By |2016-11-04T09:26:18+10:00October 17th, 2016|Tags: , , , , |0 Comments

Differential Proof A = A0ekt $$\frac{dA}{dt}=kA$$ worked examples  Show that A = A0ekt, where A0 is the initial population is a solution to the equation $$\frac{dA}{dt}=kA$$.  $$\frac{dA}{dt}=k\times A_oe^{kt}$$ since integrating this is beyond the scope of the course, we differentiate and show that the differential is right, so it must mean [...]

19 08, 2015

Projectile Motion

By |2016-11-24T09:00:08+10:00August 19th, 2015|Tags: , , , |0 Comments

Projectile Motion worked examples A cannon ball is fired at 80ms-1 at an angle of 45° to the horizontal. Calculate the height which the ball hits a vertical cliff 150m away.   draw an initial velocity diagram find the equations of motion   $$\dot x= 40\sqrt2$$ $$\ddot x [...]

19 08, 2015

Differential Equations and SHM

By |2016-11-11T09:35:14+10:00August 19th, 2015|Tags: , , , , |0 Comments

Simple Harmonic Motion: Differential Equations $$a=-n^2x$$ $$v^2=n^2(a^2-x^2)$$ $$\ddot x=a\cos nt$$ $$T=\frac{2\pi}{n}$$     $$F=\frac{1}{T}$$ worked examples Solve the differential equation $$\frac{d^2x}{dt^2}+16x=0$$ subject to the conditions x = 3 and  $$\frac{dx}{dt}=16$$ when t = 0. Find the maximum speed if x metres is the displacement of a particle moving in a [...]

19 08, 2015

Tides

By |2016-11-11T09:36:48+10:00August 19th, 2015|Tags: , , , , |0 Comments

Simple Harmonic Motion: Tides $$a=-n^2x$$ $$v^2=n^2(a^2-x^2)$$ $$\ddot x=a\cos nt$$ $$T=\frac{2\pi}{n}$$     $$F=\frac{1}{T}$$ worked examples High tide occurs at 5 am and is 9m deep.  Low tide is at 11:20 am and is 3m deep.  Find the latest time before noon a ship can enter a harbour if a minimum [...]

19 08, 2015

Simple Harmonic Motion

By |2018-05-25T19:37:23+10:00August 19th, 2015|Tags: , , , |0 Comments

Simple Harmonic Motion Simple Harmonic Motion $$a=-n^2x$$ where a is acceleration $$v^2=n^2(a^2-x^2)$$ where a is amplitude $$\ddot x=a\cos nt$$ where a is amplitude   Simple Harmonic Motion $$a = \frac{d}{{dx}}\left( {\frac{1}{2}{v^2}} \right)$$ $$T=\frac{2\pi}{n}$$   $$F=\frac{1}{T}$$ worked examples The speed v m/s of a [...]

19 08, 2015

Acceleration Proof

By |2016-11-11T09:40:00+10:00August 19th, 2015|Tags: , , , , , |0 Comments

Acceleration Proof $$a=\frac{d}{dx}(\frac{1}{2}v^2)$$ If we have acceleration as a function of position (ie in terms of x instead of t) then we cannot just integrate (since integration is in respect to t, not x) Here is a step by step explanation of the proof $$\frac{d^2x}{dt^2}=\frac{dv}{dt}$$     a = $$\frac{d^2x}{dt^2}$$ [...]

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