Alan

acceleration, a = (16 - 8)/10 m/s² = 0.8 m/s².

Use v² = u² + 2as where v = final velocity, u = initial velocity, a = acceleration, s = distance.

Rearrange as:

s = (v² - u²)/(2a)

so

s = (16² - 8²)/(2*0.8) m

$${\mathtt{s}} = {\frac{\left({{\mathtt{16}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{{\mathtt{8}}}^{{\mathtt{2}}}\right)}{{\mathtt{1.6}}}} \Rightarrow {\mathtt{s}} = {\mathtt{120}}$$

s = 120m

Check:

You could also use s = ut + (1/2)at² where t is time.

s = 8*10 + (1/2)*0.8*10²m

$${\mathtt{s}} = {\mathtt{8}}{\mathtt{\,\times\,}}{\mathtt{10}}{\mathtt{\,\small\textbf+\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right){\mathtt{\,\times\,}}{\mathtt{0.8}}{\mathtt{\,\times\,}}{{\mathtt{10}}}^{{\mathtt{2}}} \Rightarrow {\mathtt{s}} = {\mathtt{120}}$$

s = 120m

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Approximate value obtained as follows:

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Plot the real parts against the imaginary parts:

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In a spherical shell of thickness ds at a radius s, the approximate volume, dV is

$$dV=4\pi s^2ds$$

Hence the mass, dM in this shell is given by ρ*dV or:

$$dM=4\pi \frac{s^2}{s(1+s)^3}ds$$

Integrating this from 0 to r we get:

$$M(r)=4\pi \int_0^r \frac{s^2}{s(1+s)^3}ds=2\pi \frac{r^2}{(r+1)^2}$$

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4x - 3y = 11    ...(1)

10x + 2y = -1 ...(2)

Multiply the first equation by 2 and the second by 3 so that there are the same number of y's in each:

8x - 6y = 22    ...(3)

30x + 6y = -3  ...(4)

Now add (3) and (4) term by term:

38x + 0 = 19

or

38x = 19

or x = 1/2

Put this back into equation (1)

4*(1/2) - 3y = 11

or

2 - 3y = 11

Subtract 2 from both sides:

-3y = 9

Divide both sides by -3

y = -3

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It's a definite integral so:

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Apply Newton's 2nd law of motion (Force = mass*acceleration):

Vertical

Net force = -mg

-mg = m*d²y/dt²

so the mass cancels out and we are left with:

d²y/dt² = -g  which is the same as your equation Melody.

Horizontal

Net force = 0

0 = m*d²x/dt²

so dividing by the mass we are left with

d²x/dt² = 0

Mass is not relevant here.

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If, by "twenties" you are referring to twenty-pence pieces, then, as there are 5 of them in £1, there are 9*5 = 45 in £9.

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