  Let $SL_2$ be the rank one simple algebraic group defined over an
algebraically closed field $k$ of characteristic $p>0$. The paper presents a
new method for computing the dimension of the cohomology spaces
$\text{H}^n(SL_2,V(m))$ for Weyl $SL_2$-modules $V(m)$. We provide a closed
formula for $\text{dim}\text{H}^n(SL_2,V(m))$ when $n\le 2p-3$ and show that
this dimension is bounded by the $(n+1)$-th Fibonacci number. This formula is
then used to compute $\text{dim}\text{H}^n(SL_2, V(m))$ for $n=1, 2,$ or $3$.
For $n>2p-3$, an exponential bound, only depending on $n$, is obtained for
$\text{dim}\text{H}^n(SL_2,V(m))$. Analogous results are also established for
the extension spaces $\text{Ext}^n_{SL_2}(V(m_2),V(m_1))$ between Weyl modules
$V(m_1)$ and $V(m_2)$. In particular, we determine the degree three extensions
for all Weyl modules of $SL_2$. As a byproduct, our results and techniques give
explicit upper bounds for the dimensions of the cohomology of the Specht
modules of symmetric groups, the cohomology of simple modules of $SL_2$, and
the finite group of Lie type $SL_2(p^s)$.
