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Theoretical Model

The theoretical model of d can be found by calculating the time t taken for the object to travel through the air.

The force of the object down the slope: F = ma.

F = mgsin q mgsin q can be replaced for F in F = ma.

mgsin q = ma mgsin q = ma The mass of the object cancels out.

gsin q = a

To find t, v must be found first. v2 = u2 + 2as is the equation linking acceleration, distance and velocity together.

v2 = u2 + 2as As a been found and s = l, the known variable, these can be placed in the equation.

v2 = u2 + 2glsin q There is no starting velocity, so u2 = 0

v2 = 0 + 2glsin q

v2 = 2glsin q

v = The velocity of the object is known.

As the object is sent over the edge of the slide, it moves in a vertical and horizontal direction. To simplify calculations, positive numbers will be used where possible. Vertically, down will be regarded as a positive direction, the same direction as h values; horizontally, the same direction as d values will be used.

Using s = ut + at2 where the components of u are:
 v sin q and > cos q .

 v Vertically h = vtsin q + gt2

 > Horizontally d = vtcos q

 

Time t can be found with h being inserted into d.

h = vtsin q + gt2

vtsin q + gt2 h = 0 Treat as a quadratic equation:

t = t =

Now that t has been found it can be inserted into d (vtcos q ) to give theoretical results for fixed values of q and changing values of l.

d = vcos q

As there are a large number of calculations it would save time if d was found by an automated method. The following tablulated results of theoretical d were found using MS Excel, a spreadsheet.

 

The MS Excel formulae used for q = 24 and l = 0.1; where A2 = l, G2 = v, H2 = h, I2 = t, 9.8 = g, are as follows:

Velocity v

=SQRT(2*9.8*A2*SIN(RADIANS(24)))

Time t

=(-1*G2*SIN(RADIANS(24))+ SQRT((POWER(G2,2))
*POWER(SIN(RADIANS(24)),2)+ 2*9.8*H2))/9.8

Theoretical d

=100*G2*I2*COS(RADIANS(24))

The radian instructions convert degrees from the default radian angles. In the theoretical formula the answer is normally in metres, but in this case the answer is multiplied by 100 to give it in centimetres for comparison with the experimental data. There is one reason why the experimental data was taken in centimetres, it is the scale used on the rulers where the ratchet socket lands on the floor. This will have no negative effect on any formulae as values of d can be easily converted back into metres and vice versa with no reduction of quality.

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