PROBLEM:
Calculate the minimum ejection velocity with which a shell must be fired to strike a target 1000ft high and directly overhead.
QUESTIONS:
I use integration to work back from 32ft/sec and arrive at the two expressions.
(Vi) is initial velocity
(t) is in seconds
I've tried solving for (t) in the expression for velocity. I substituted it in the expression for (s) displacement. I then tried to set it equal to zero and solved for (Vi) initial velocity.
I came out with 179.38614. However, the correct answer is shown as 80[sqrt(10)]ft./sec which is approximately 252.98 rounded off.
I don't understand where I'm going wrong. Should I solve for (t) instead of (Vi)?
Your algebra is correct (though I'm not crazy about the notation...). You have an arithmetic mistake: your equation correctly reduces to
$$\frac{1}{32} \cdot v_i^2 \ - \ \frac{16}{32^2} \cdot v_i^2 \ = 1000 \ , $$
which should produce $\frac{1}{32} \cdot v_i^2 \ - \ \frac{1}{2 \ \cdot \ 32} \cdot v_i^2 \ = 1000 \ \Rightarrow \ \frac{1}{64} \cdot v_i^2 \ = 1000$ . This gives the indicated answer.