A ball of mass $0.45 \mathrm{kg}$ which is initially at rest is hit by a bat. The bat remains in contact with the ball for $3\times {10}^{-3} \mathrm{s}$. During this time period the force on the ball by the bat is given as $F\left(t\right)=\left[\left(\alpha \times {10}^6\right)t-\left(\beta \times {10}^9\right)t^2\right] \mathrm{N}$ where $\alpha$ and $\beta$ are constants. The ball's speed immediately as it loses contact with the bat is $20 \mathrm{m} {\mathrm{s}}^{-1}$. The correct relation between $\alpha$ and $\beta$ is