A projectile of mass m is launched with an initial velocity vecvi making an angle θ with the horizontal as shown in figure. The projectile moves in the gravitational field of the Earth. The angular momentum of the projectile about the origin when it is at the highest point of its trajectory is

Question

A projectile of mass m is launched with an initial velocity vecvi making an angle θ with the horizontal as shown in figure. The projectile moves in the gravitational field of the Earth. The angular momentum of the projectile about the origin when it is at the highest point of its trajectory is (A) (-m vi3 sin θ2 cos θ/2 g) hatk (B) (m vi3 sin θ2 cos θ/2 g) hatk (C) (-m vi3 sin θ2 cos θ/g) hatk (D) (2 m vi3 sin θ2 cos θ/g) hatk

Tardigrade Admin · Accepted Answer

At the highest point of the trajectory,

x = \frac{1}{2} R = \frac{v _{i}^{2} s i n 2 θ}{2 g} and y = h_{m a x} = \frac{( v _{i} s i n θ ) ^{2}}{2 g}

The angular momentum is then

L_{1} = r_{1} \times m v_{1}

= [\frac{v _{i}^{2} s i n 2 θ}{2 g} \hat{i} + \frac{( v _{i} s i n θ ) ^{2}}{2 g} \hat{j}] \times m v_{x i} \hat{i}

= \frac{- m v _{i}^{3} s i n θ ^{2} c o s θ}{2 g} \hat{k}

$\frac{- m v _{i}^{3} s i n θ ^{2} c o s θ}{2 g} \hat{k}$

$\frac{m v _{i}^{3} s i n θ ^{2} c o s θ}{2 g} \hat{k}$

$\frac{- m v _{i}^{3} s i n θ ^{2} c o s θ}{g} \hat{k}$

$\frac{2 m v _{i}^{3} s i n θ ^{2} c o s θ}{g} \hat{k}$

Q. A projectile of mass m is launched with an initial velocity vi​ making an angle θ with the horizontal as shown in figure. The projectile moves in the gravitational field of the Earth. The angular momentum of the projectile about the origin when it is at the highest point of its trajectory is

2g−mvi3​sinθ2cosθ​k^

2gmvi3​sinθ2cosθ​k^

g−mvi3​sinθ2cosθ​k^

g2mvi3​sinθ2cosθ​k^

At the highest point of the trajectory, x=21​R=2gvi2​sin2θ​ and y=hmax​=2g(vi​sinθ)2​ The angular momentum is then L1​=r1​×mv1​ =[2gvi2​sin2θ​i^+2g(vi​sinθ)2​j^​]×mvxi​i^ =2g−mvi3​sinθ2cosθ​k^

$\frac{- m v _{i}^{3} s i n θ ^{2} c o s θ}{2 g} \hat{k}$

$\frac{m v _{i}^{3} s i n θ ^{2} c o s θ}{2 g} \hat{k}$

$\frac{- m v _{i}^{3} s i n θ ^{2} c o s θ}{g} \hat{k}$

$\frac{2 m v _{i}^{3} s i n θ ^{2} c o s θ}{g} \hat{k}$