Question

A small particle of mass `m` is projected at an angle `theta` with x-axis with initial velocity `upsilon_(0)` in x-y plane as shown in Fig. Calculate the momentum of the particle
at `t lt (upsilon_(0) sin theta)/(g)`.

Answer 1

As time of fight `T = (2 upsilon_(0) sin theta)/(g)`,
therefore, at `t lt (upsilon_(0) sin theta)/(g)`, particle has not reached the maximum height. Let `overset rarr(r )` be position vector the particle as shown in Fig.

`overset rarr(r ) = overset rarr(u) t - (1)/(2) overset rarr(g) t^(2)`
`= (upsilon_(0) cos theta hati + upsilon_(0) sin theta hatj) t - (1)/(2) gt^(2) hatj`
`overset rarr(r ) = upsilon_(0) cos theta (t) hati + (upsilon_(0) sin theta(t) - (1)/(2) gt^(2))hatj`
and `overset rarr(upsilon) = overset rarr(u) - overset rarr(g) t`
`= upsilon cos theta hati + upsilon theta hatj - gt hatj`
`= upsilon_(0) cos theta hati + (upsilon_(0) sin theta - gt) hatj`
Angular momentum of particle,
`overset rarr(L) = m(overset rarr(r ) xx overset rarr(upsilon))` ..(i)
Now,
`overset rarr(r ) xx overset rarr(upsilon)|[hati,hatj, hatk],[upsilon_(0)cos theta(t), upsilon_(0) sin theta(t)-(1)/(2) g t^(2),0],[upsilon_(0)cos theta, (upsilon_(0)sin theta-g t),0]|`
`= hati (0 -0) - hatj (0 - 0)`
`+ hatk [upsilon_(0)^(2) cos theta sin theta(t) - upsilon_(0) cos theta g t^(2)`
`- upsilon_(0)^(2) cos theta sin theta t+ (1)/(2) upsilon_(0) cos theta g t^(2)]`
`overset rarr(r ) xx overset rarr(upsilon) = - (1)/(2) upsilon_(0) cos theta g t^(2) hatk`
From (i),
`overset rarr(L) = - (1)/(2) m g upsilon_(0) t^(2) cos theta hatk`