A particle of mass m is moving in time t on a trajectory given by,...

\overset{\to}{r} = 10\alpha{t}^{2}\overset{^}{i} + 5\beta\left(t - 5\right)\overset{^}{j}

\overset{\to}{v} = 20\alpha + \overset{^}{i} + 5\beta\overset{^}{j}

$\overset{\to}{L} = m\left(\overset{\to}{r} \times \overset{\to}{v}\right)$

= m\left[10\alpha{t}^{2}\overset{^}{i} + 5\beta\left(t - 5\right)\overset{^}{j}\right] \times \left[20\alpha\overset{^}{i} + 5\beta\overset{^}{j}\right]

\overset{\to}{L} = m\left[50\alpha\beta{t}^{2}\overset{^}{k} - 100\alpha\beta\left({t}^{2} - 5t\right)\overset{^}{k}\right]

At $t = 0, \overset{\to}{L} = \overset{\to}{0}$

$50\alpha\beta{t}^{2} - 100\alpha\beta\left({t}^{2} - 5t\right) = 0$

$t - 2\left(t - 5\right) = 0$

$t = 10sec$