(A) Let the horizontal direction be the $x$-axis and the vertical direction be the $y$-axis.At time $t$,the velocity vector of the first particle is $

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(A) Let the horizontal direction be the $x$-axis and the vertical direction be the $y$-axis.At time $t$,the velocity vector of the first particle is $\vec{v}_1 = u_1 \hat{i} - gt \hat{j}$.The velocity vector of the second particle is $\vec{v}_2 = -u_2 \hat{i} - gt \hat{j}$.Since the velocities are mutually perpendicular,their dot product must be zero: $\vec{v}_1 \cdot \vec{v}_2 = 0$.$(u_1 \hat{i} - gt \hat{j}) \cdot (-u_2 \hat{i} - gt \hat{j}) = 0$.$-u_1 u_2 + g^2 t^2 = 0$.$g^2 t^2 = u_1 u_2$.$t = \frac{\sqrt{u_1 u_2}}{g}$.

Class 11 Physics 3-2.Motion in Plane Horizontal Projectile Motion

Difficult

Two particles are simultaneously projected in opposite directions horizontally from a given point in space,where gravity $g$ is uniform. If $u_1$ and $u_2$ are their initial speeds,then the time $t$ after which their velocities are mutually perpendicular is given by

A
$\frac{\sqrt{u_1 u_2}}{g}$
B
$\frac{\sqrt{u_1^2+u_2^2}}{g}$
C
$\frac{\sqrt{u_1(u_1+u_2)}}{g}$
D
$\frac{\sqrt{u_2(u_1+u_2)}}{g}$

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3-2.Motion in Plane

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Horizontal Projectile Motion

The path of a projectile in the absence of air drag is shown in the figure by a dotted line. If air resistance is not ignored,then which one of the paths shown in the figure is appropriate for the projectile?

Easy

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Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$.
Assertion $A$: When a body is projected at an angle $45^{\circ}$,its range is maximum.
Reason $R$: For maximum range,the value of $\sin 2\theta$ should be equal to one.
In the light of the above statements,choose the correct answer from the options given below:

MediumJEE MAIN 2023

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$A$ ball is thrown at $30^{\circ}$ with the horizontal from the top of a roof $20 \,m$ high with a speed of $13 \,ms^{-1}$. At what distance from the throwing point will the ball, once again, be at a height of $20 \,m$ from the ground (in $\,m$)? $(g = 10 \,ms^{-2})$

MediumAP EAMCET 2021

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$A$ boy playing on the roof of a $10\, m$ high building throws a ball with a speed of $10\, m/s$ at an angle of $30^o$ with the horizontal. How far from the throwing point will the ball be at the height of $10\, m$ from the ground? $\left[ g = 10\, m/s^2, \sin 30^o = \frac{1}{2}, \cos 30^o = \frac{\sqrt{3}}{2} \right]$

Medium

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$A$ projectile is thrown at a speed which is twice its speed at its maximum height. If $R$ and $H$ are its range and maximum height respectively,then the ratio $\frac{R}{H}$ is

EasyTS EAMCET 2021

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Two particles are simultaneously projected in opposite directions horizontally from a given point in space,where gravity $g$ is uniform. If $u_1$ and $u_2$ are their initial speeds,then the time $t$ after which their velocities are mutually perpendicular is given by

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The path of a projectile in the absence of air drag is shown in the figure by a dotted line. If air resistance is not ignored,then which one of the paths shown in the figure is appropriate for the projectile?

$A$ ball is thrown at $30^{\circ}$ with the horizontal from the top of a roof $20 \,m$ high with a speed of $13 \,ms^{-1}$. At what distance from the throwing point will the ball, once again, be at a height of $20 \,m$ from the ground (in $\,m$)? $(g = 10 \,ms^{-2})$

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