2003 Paper 1 Q9

Year: 2003
Paper: 1
Question Number: 9

Course: LFM Pure and Mechanics
Section: Projectiles

Difficulty: 1500.0 Banger: 1500.0

projectiles quadratic in tan trajectory equation discriminant inequality sum of angles trigonometric identity optimisation

Problem

A particle is projected with speed \(V\) at an angle \(\theta\) above the horizontal. The particle passes through the point \(P\) which is a horizontal distance \(d\) and a vertical distance \(h\) from the point of projection. Show that \[ T^2 -2kT + \frac{2kh}{d}+1=0\;, \] where \(T=\tan\theta\) and \(\ds k= \frac{V^2}{gd}\,\). %Derive an equation relating \(\tan \theta\), \(V\), \(g\), \(d\) and \(h\). Show that, if \(\displaystyle {kd > h + \sqrt {h^2 + d^2}}\;\), there are two distinct possible angles of projection. Let these two angles be \(\alpha\) and \(\beta\). Show that \(\displaystyle \alpha + \beta = \pi - \arctan ( {d/ h}) \,\).

No solution available for this problem.

Rating Information

Difficulty Rating: 1500.0

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Banger Rating: 1500.0

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Problem source

A particle is projected with speed  $V$ 
at an angle $\theta$ above  the horizontal. 
The particle passes through the  point $P$ which is a horizontal distance
$d$ and a vertical distance
$h$  from the point of projection. 
Show that 
\[
T^2 -2kT + \frac{2kh}{d}+1=0\;,
\]
where $T=\tan\theta$ and $\ds k= \frac{V^2}{gd}\,$.
%Derive an equation relating $\tan \theta$, $V$, $g$, $d$ and $h$. 
Show that, if 
$\displaystyle {kd > h + \sqrt {h^2 + d^2}}\;$,
 there are two distinct possible angles of projection.

Let these two angles be $\alpha$ and $\beta$. 
Show that 
$\displaystyle \alpha + \beta = \pi - \arctan ( {d/  h}) \,$.

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