Question

A particle moving on a curve has the position given by $x=f^{\prime }\left(t\right)\sin t+f^{\prime \prime }\left(t\right)\cos t,y=f^{\prime }\left(t\right)\cos t-f^{\prime \prime }\left(t\right)\sin t$ at time $t$ where $f$ is a thrice-differentiable function.Then the velocity of the particle at time $t$ is

• A. $f^{\prime \prime \prime }\left(t\right)$
• B. $f^{\prime }\left(t\right)+f^{\prime \prime \prime }\left(t\right)$
• C. $f^{\prime }\left(t\right)+f^{\prime \prime }\left(t\right)$
• D. $f^{\prime }\left(t\right)-f^{\prime \prime \prime }\left(t\right)$

Answer 1

The correct option is B $f^{\prime }\left(t\right)+f^{\prime \prime \prime }\left(t\right)$

$V_x$
$=\frac{dx}{dt}$
$=f^{\prime }\left(t\right)cost+f"\left(t\right)sint-f"\left(t\right)sint+f{"}^{\prime }\left(t\right)cost$
$=cost\left(f^{\prime }\right(t)+f^{\prime \prime \prime }(t\left)\right)$ ...(i)
$V_y$
$=\frac{dy}{dt}$
$=-f^{\prime }\left(t\right)sint+f"\left(t\right)cost-f"\left(t\right)cost-f{"}^{\prime }\left(t\right)sint$
$=-sint\left(f^{\prime }\right(t)+f^{\prime \prime \prime }(t\left)\right)$ ....(ii)
Hence
Net velocity
$V_{net}$
$=\sqrt{V_x^2+V_y^2}$
$=\left(f^{\prime }\right(t)+f^{\prime \prime \prime }(t\left)\right)\sqrt{si{n}^{2}t+co{s}^{2}t}$
$=f^{\prime }\left(t\right)+f^{\prime \prime \prime }\left(t\right)$.