smc-1062-98-Vectors

A particle which is projected from the origin with initial speed `u` ms^-1 at an angle of `theta` to the positive `x`-axis lands on the `x`-axis, as shown in the diagram. The particle is subject to an acceleration due to gravity of `g` ms^-1.

The position vector of the particle, `underset~r (t)`, where `t` is the time in seconds after the particle is projected, is given by

`underset~r (t) = ((ut cos theta),( - {g t^2}/{2} + u t sin theta))`. (Do NOT prove this.)

For some value(s) of `theta` there will be two times during the time of flight when the particle’s position vector is perpendicular to its velocity vector.

Find the value(s) of `theta` for which this occurs, justifying that both times occur during the time of flight. (5 marks)

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`text{See Worked Solutions}`

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`underset~r (t) = ((ut cos theta),( – {g t^2}/{2} + u t sin theta)) , \ underset~v (t) = ((u cos theta),(-g t + u sin theta))`

♦♦ Mean mark 31%.

`text{If} \ \ underset~r (t) ⊥ underset~v (t) \ => \ underset~r (t) * underset~v (t) = 0`

`u t cos theta * u cos theta + (-{g t^2}/{2} + u t sin theta) (- g t + u sin theta)`	`= 0`
`u^2 t cos^2 theta + {g^2 t^3}/{2} – {g t^2}/{2} u sin theta – g t^2 u sin theta + u^2 t sin theta`	`= 0`
`u^2 t (cos^2 theta + sin^2 theta) + {g^2 t^3}/{2} – {3 u g t^2}/{2} sin theta`	`= 0`
`u^2 t + {g^2 t^3}/{2} – {3 u g t}/{2} sin theta`	`= 0`
`g^2 t^2 – 3 u g t sin theta + 2 u^2`	`= 0`

`text{If 2 roots} , \ Δ > 0 :`

`(-3 u g sin theta)^2 – 4 g^2 \ 2 u^2`	`> 0`
`9 u^2 g^2 sin^2 theta -8 g^2 u^2`	`> 0`
`u^2 g^2 (9 sin^2 theta – 8)`	`> 0`
`sin^2 theta`	`> 8/9`
`theta`	`> sin^-1 ({2 sqrt2}/{3})`

`text{S} text{ince} \ \ 0 < theta < pi/2 \ , text{only one value of} \ theta \ text{satisfies}`

`text{Checking valid times of flight}\ (t_f):`

`1/2 \ text{time of flight} \ => underset~j text{-component of} \ underset~v (t) = 0`

`1/2 g t_f`	`= u sin theta`
`t_f`	`= {2 u sin theta}/{g}`

`t`	`={3 u g sin theta ± sqrt{9 u^2 g^2 sin^2 theta – 4g^2 * 2 u ^2}}/{2 g^2}`
	`= 1/2 ({3 u sin theta}/{g} ± {u sqrt{9 sin^2 theta -8}}/{g})`
	`= {u}/{2g} (3 sin theta ± sqrt{9 sin^2 theta – 8})`

`text{S} text{ince} \ sqrt{9 sin^2 theta – 8} < sqrt{9 sin^2 theta} < 3 sin theta`

`=> \ text{in both solutions} \ \ t > 0`

`text{Consider the longer time:}`

`{u}/{2g} (3 sin theta + sqrt{9 sin^2 theta – 8})`

`< {u}/{2g} (3 sin theta + sqrt{9 sin^2 theta – 8 sin^2 theta})`

`< {u}/{2g} (3 sin theta + sin theta)`

`< {2 u sin theta}/{g}`

`:. \ text{Longer time < time of flight}`

`:. \ text{Both times occur during time of flight.}`