2003 Paper 2 Q11

Year: 2003
Paper: 2
Question Number: 11

Course: LFM Pure and Mechanics
Section: Projectiles

Difficulty: 1600.0 Banger: 1488.4

projectiles flight time direction of motion trigonometric quadratic equation velocity components angle of elevation

Problem

A particle $P_1$ is projected with speed $V$ at an angle of elevation ${\alpha}\,\,\,( > 45^{\circ})\,,\,\,\,$ from a point in a horizontal plane. Find $T_1$, the flight time of $P_1$, in terms of ${\alpha}, V \hbox{ and } g\,$. Show that the time after projection at which the direction of motion of $P_1$ first makes an angle of $45^{\circ}$ with the horizontal is $\frac12 (1-\cot \alpha)T_1\,$. A particle $P_2$ is projected under the same conditions. When the direction of the motion of $P_2$ first makes an angle of $45^{\circ}$ with the horizontal, the speed of $P_2$ is instantaneously doubled. If $T_2$ is the total flight time of $P_2$, show that $$ \frac{2T_2}{T_1} = 1+\cot{\alpha} +\sqrt{1+3\cot^2{\alpha}} \;. $$

No solution available for this problem.

Rating Information

Difficulty Rating: 1600.0

Difficulty Comparisons: 0

Banger Rating: 1488.4

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Problem source

A particle $P_1$ is projected with speed $V$ at an angle of elevation
${\alpha}\,\,\,( > 45^{\circ})\,,\,\,\,$ 
from a point in a horizontal plane. 
Find $T_1$, the flight time of $P_1$, in terms of
${\alpha}, V \hbox{    and    } g\,$.
Show that the time after projection at 
which the direction of motion of $P_1$ first 
makes an angle of 
$45^{\circ}$ with the horizontal is $\frac12 (1-\cot \alpha)T_1\,$.

A particle $P_2$  is projected 
under the same conditions.
When the direction of the motion of $P_2$ 
first makes an angle of  $45^{\circ}$ with the horizontal, the speed of
$P_2$ is instantaneously doubled. If $T_2$ is the total flight time of
$P_2$,  show that 
$$
\frac{2T_2}{T_1}
=  1+\cot{\alpha}
+\sqrt{1+3\cot^2{\alpha}} \;.
$$

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