Two particles are projected in air with speed `v_(0)` at angles `theta_(1)` and `theta_(2)` (both acute) to the horizontal,respectively.If the height reached by the first particle greater than that of the second,then thick the right choices

Options

A. Angle of projection `q_(1) gt q_(2)`
B. Time of flight `T_(1) gt T_(2)`
C. Horizontal range `R_(1) gt R_(2)`
D. Total energy `U_(1) gt U_(2)`

Answer

A, B, C

Explanation

We know that maximum height reached by a projectile,
`H=(u^(2)sin^(2) theta)/(2g)`
`H_(1)=(v_(0)^(2)sin^(2)theta_(1))/(2g) " "`(for first particle)
`H_(2)=(v_(0)^(2)sin^(2)theta_(2))/(2g)" " `(for second particle)
According to question, we know that
`H_(1)gt H_(2)`
`rArr " " (v_(0)^(2)sin^(2)theta_(1))/(2g) gt(v_(0)^(2)sin^(2)theta_(2))/(2g)`
`rArr " " sin^(2)theta_(1) gt sin^(2) theta_(2)`
`rArr " " sin^(2)theta_(1)-sin^(2)theta_(2) gt0`
`rArr " " (sintheta_(1)-sin theta_(2))(Sin theta_(1)+sin theta_(2)) gt =0`
Thus, either `" "sin theta_(1)+sin theta_(2) gt 0`
`rArr " " sin theta_(1) -sin theta_(2) gt 0`
`rArr " " sin theta_(1) gt sin theta_(2) or theta_(1) gt theta_(2)`
Time of fight, `" " T=(2usin theta)/(g)=(2v_(0)sin theta)/(g)`
Thus, `" T_(1)=(2v_(0)sin theta_(1))/(g)`
`T_(2)=(2 v_(0)sin theta_(2))/(g)`
(Here , `T_(1)`=Time of flight particle and `T_(2)` = Time of flight of second particle).
As, `" " sin theta_(1) gt sin theta_(2)`
Hence, `" " T_(1) gt T_(2)`
We know that
Range , `R=(u^(2)sin 2 theta)/(g)=(v_(0)^(2)sin 2 theta)/(g)`
`R_(1)`=Range of first particle `=(u_(0)^(2)sin 2 theta_(1))/(g)`
`R_(2)`=Range of second particle `=(v_(0)^(2)sin 2 theta_(2))/(g)`
Given, `" " sin theta_(1) gt sin theta_(2)`
`rArr " " sin 2 theta_(1) gt sin 2 theta_(2)`
`rArr " " (R_(1))/(R_(2))=(sin 2 theta_(1))/(sin 2 theta_(2)) gt 1`
`rArr " " R_(1) gt R_(2)`
Total energy for the first particle,
`U_(1)=KE+PE=(1)/(2)m_(1)v_(0)^(2)`
`" "` (This value will be constant throughout the journey)
`U_(2)=KE+PE=(1)/(2) m_(2)v_(0)^(2) " "` (Total energy for the second particle)
Total energy for the second particle
If `" " m_(1)=m_(2) ` then `U_(1)=U_(2)`
`" " m_(1) gt m_(2)` then `U_(1) gt U_(2)`
`" " m_(1) lt m_(2)`, then `U_(1) lt U_(2)`