Question

A rock is thrown upward from level ground in such a way that the maximum height of its flight is equal to its horizontal range R.

(a) At what angle theta is the rock thrown?

(b) In terms of its original range R, what is the range R_max the rock can attain if it is launched at the same speed but at the optimal angle for maximum range?

(c) Would your answer to part (a) be different?

Answer 1

the range equation is R=v0^2 sin(2t)/g where I use t for theta

so, take the ratio of Rmax to R:

Rmax/R = v0^2 sin(2*45)/g / v0^2sin(2t)/g

the v0 and g terms cancel, sin 90=1, leaving

Rmax/R = 1/sin(2t) or Rmax = R/sin(2t)

that would help

Answer 2

(a) Θ =
`75.96^(0)`

(b)
`R_(max)` is achieved when T =
`45^(0)`, thus
`R_(max)` = R ÷ Sin2T

(c) Yes

Step-by-step explanation:

(a) Max. height (
`H_(max)`) =
`(U^(2)Sin^(2)T)/(2g)`

Range (R) =
`(U^(2)Sin 2T )/(g)`

where T represents theta

So for, R =
`H_(max)`

⇒
`(U^(2)Sin 2T )/(g)` =
`(U^(2)Sin^(2)T)/(2g)`

Using trigonometric functions,

`Sin^(2)T` = 2SinTCosT =
`(Sin^(2)T )/(2)`

⇒ 4 = Tan T

T =
`Tan^(-1) 4`

=
`75.96^(0)`

∴ Theta =
`75.96^(0)`

(b) Since, R =
`(U^(2)Sin 2T )/(g)`

`R_(max)` is achieved when T =
`45^(0)`

`R_(max)` =
`(U^(2)Sin 2T )/(g)`

∴
`R_(max)` = R ÷ Sin2T