1. A stone is thrown vertically upwards with an initial velocity $V$. Neglecting air resistance and assuming the acceleration due to gravity is $10 \mathrm{~m} / \mathrm{c}^{2}$, determine the values of $V$ for which all moments of reaching a height of 10 m will lie between: A) the first and second seconds after the start of the motion; B) the second and fourth seconds after the start of the motion.
Solution. The dependence of height on time is $h(t)=V t-\frac{g t^{2}}{2}$. Therefore, the stone will be at a height of 10 m at the moments of time $10=V t-5 t^{2}$. This results in the equation $5 t^{2}-V t+10=0$, which for $V^{2} \geqslant 200$ (i.e., for $V \geqslant 10 \sqrt{2}$) has roots: $t_{1,2}=\frac{V \pm \sqrt{V^{2}-200}}{10}$.

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& \text { A) } 12 \text{ and } t_{2}>2 \text{ cannot be satisfied simultaneously.}

Answer: $A) V \in[10 \sqrt{2} ; 15) \text{ m/s };$ B) $V \in \varnothing$.

Criteria: 20 points - correct (not necessarily the same as above) solution and correct answer (including the value $V=15$ m/s in the answer does not affect the score); 15 points - any significant errors (with correct answers); 10 points - correct solution of one of the parts: either A or B; 5 points - the correct system of irrational inequalities is written, but errors are made in solving it; $\mathbf{0}$ points all other cases.
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