Answer
If the train travels at a speed of at least $0.6~c$, then according to experimenters on the ground, the whole train will fit inside the tunnel at one time.
Work Step by Step
As the train moves with a certain speed relative to the observers, the observers will see that the length of the train decreases due to length contraction.
We can find the required speed:
$L = L_0~\sqrt{1-\frac{v^2}{c^2}}$
$0.80~L_0 = L_0~\sqrt{1-\frac{v^2}{c^2}}$
$0.80 = \sqrt{1-\frac{v^2}{c^2}}$
$(0.80)^2 = 1-\frac{v^2}{c^2}$
$\frac{v^2}{c^2} = 1 - (0.80)^2$
$v^2 = [1 - (0.80)^2~]~c^2$
$v = \sqrt{1 - (0.80)^2}~c$
$v = 0.6~c$
If the train travels at a speed of at least $0.6~c$, then according to experimenters on the ground, the whole train will fit inside the tunnel at one time.
