Answer
The least speed is $~~42.0~m/s$
Work Step by Step
From the graph, we can see that the range is 240 m when $\theta = 45^{\circ}$
We can find $v_0$:
$R = \frac{v_0^2~sin~2\theta}{g}$
$v_0^2 = \frac{gR}{sin~2\theta}$
$v_0 = \sqrt{\frac{gR}{sin~2\theta}}$
$v_0 = \sqrt{\frac{(9.80~m/s^2)(240~m)}{sin~(2)(45^{\circ})}}$
$v_0 = 48.5~m/s$
We can find an expression for the flight time $t$:
$t = \frac{2~v_0~sin~\theta}{g}$
$t_{max}$ will occur when $\theta = 90^{\circ}$
$0.500~t_{max}$ will occur when $sin~\theta = 0.500$, which occurs when $\theta = 30^{\circ}$
The least speed occurs at maximum height when the speed is $v_x$
We can find $v_x$ when $\theta = 30^{\circ}$:
$v_x = (48.5~m/s)~cos~30^{\circ} = 42.0~m/s$
The least speed is $~~42.0~m/s$
