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$MEC{N}_2{O}_{13}$

QUESTION $1(15($:$m^{m-c}\}$

Figure $1$: Particle following a brachistochrone curve

A particle $P$ is moving along a path which is given by the parametric equations below for the $x$ and $y$ co$-$ordinates, where $t$ is in seconds and $x$ and $y$ are in m $.$ The curve is for the period of time between $t=0s$ and $t=4s.$

$x(t)=\frac{t}{4}-sin(\frac{t}{4})$

$y(t)=1+cos(\frac{t}{4})$

Remember to use radian mode in your calculators for this question and not for the other questions!

For the instant when $t=1\mathrm{.}5s$ :

$($a$)$ Determine the magnitude and direction of the resultant velocity of the particle $P.(6$ marks$)$

$($b$)$ Determine the magnitude of the angular velocity, ${}^{}$ of the radial position vector that points

$(9$ marks$)$ to the particle.

$($c$)$ BONUS: Determine the magnitude of the normal acceleration of the particle, $a_n.$ Hint: the radius of curvature is unknown and does not need to be calculated.

$($BONUS: $4$ marks$)$

$[$Total: $15$ marks, BONUS: $4]$

PTO$/$ Page $3...$