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Wednesday, July 2, 2014

Parabolic motion (motion in two dimension)

Question
A stone is thrown from ground level over horizontal ground. It just clears three walls, the successive distances between them being r and 2r. The inner wall is 15/7 times as high as the outer walls which are equal in height. The total horizontal range is nr, where n is an integer. Find n.
Solution
Let us just assume that both the outer walls are equal in height say \(h\) and they are at equal distance \(x\) from the end points of the parabolic trajectory as can be shown below in the figure.

Now equation of the parabola is
 \(y = bx - c{x^2}\)                                       (1)
 \(y = 0\) at \(x = nr = R\)
where \(R\) is the range of the parabola.
Putting these values in equation (1) we get
\(b = cnr\)                                                       (2)
Now the range \(R\) of the parabola is
\(R = a + r + 2r + a = nr\)
This gives
\(a = \left( {n - 3} \right)\frac{r}{2}\)               (3)
The trajectory of the stone passes through the top of the three walls whose coordinates are
\(\left( {a,h} \right),\left( {a + r,\frac{{15}}{7}h} \right),\left( {a + 3r,h} \right)\)
Using these co-ordinates in equation 1 we get
\(h = ab - c{a^2}\)                                           (4)
\(\frac{{15}}{7}h = b(a + r) - c{(a + r)^2}\)                              (5)
\(h = b(a + 3r) - c{(a + 3r)^2}\)                                      (6)
After combining (2), (3), (4), (5) and (6) and solving them we get n = 4.
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