A particle's position r$($t$),$ velocity v$($t$),$ and acceleration a$($t$)$ are functions of time t$.$

Suppose that a particle has r$($t$)=$Rcos$(\backslash$pi t$^(2))$i$+$Rsin$(\backslash$pi t$^(2))$jR is some positive constantv$($t$),$v$($t$),$hat$($v$)($t$),$a$($t$)$hat$($v$)$ simply means $($d$)/($dz$)$v$.$ In his writing, Newton places a dot over a time$-$

dependent quantity to signify its temponal derivative.cos$\_((1))$a$(1)$ and Proj$\_($v$(1))$a$(1)$v$^()(1)$ and Comp$\_($v$(1))$a$(1)($a$(1)-$Prof$\_(\backslash$gamma $(1))$a$(1))=|($a$(1)-$Proj$\_(\backslash$tau $(1))$a$(1))|,$ and $(($v$(1))^(2))/($R$).$

$($d$)$ Sketch the trajectory of the particle. At the point r$(1),$ sketch the four vectors

v$(1),$a$(1),$Proj$\_($v$(1))$a$(1),$a$(1)-$Prof$\_($v$(1))$a$(1)$;