Answer
The friend is wrong.
Work Step by Step
Let's note by $t$ (in seconds) the time needed by the cheetah to catch up with the antelope.
The distance, $C$, travelled by the cheetah running at a speed of $90$ feet per second will be $90$ times $t$. Therefore $$C = 90t. \tag{1}$$ Similarly, the distance, $A$, travelled by the antelope running at $60$ feet per second in $t$ seconds will be $60t$. However, the cheetah is behind the antelope by $650$ ft. Hence, the total distance the antelope travelled (from the origin point) is $60t + 650$. Therefore $$A = 60t + 650.\tag{2}$$ In order for the cheetah to catch up, the distances $C$ and $A$ should be equal. Therefore, we set up the equation $$C = A.\tag{3}$$
Using equations $(1)$ and $(2)$ in equation $(3)$ we have:
$$\begin{align}
90t& = 60t + 650\\
30t& = 650\\
t &\approx 21.66\text{ seconds}.
\end{align}$$ The cheetah can catch up if it runs constantly at its maximum speed for $21.66$ seconds which is physically unattainable for the cheetah as per the question, so the friend is wrong. The cheetah will need more than $21$ seconds to catch up. However the cheetah can only run at its full speed for $20$ seconds after which the speed will reduce. Hence, the antelope can get away safely.
