Variance of jointly distributed random variables
sorry but your question is immediate...and you do not need to know how $U$ is distributed. Are you sure about the question?
You are given that
$$(V|U=u)\sim \text{Exp}(u)$$
thus
$$\mathbb{V}[V|U=u]=\frac{1}{u^2}$$
Sure that you are not asked to find $\mathbb{V}[V]$?