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Question Number 188203 by moh777 last updated on 26/Feb/23

Answered by mr W last updated on 26/Feb/23

g=32.2 ft/s^2 (1) h_1 =v_1 (t+3)−((g(t+3)^2 )/2) h_2 =v_2 t−((gt^2 )/2) h_1 =h_2 =350 ft v_1 (t+3)−((g(t+3)^2 )/2)=h_1 =350 t+3=(1/g)(v_1 ±(√(v_1 ^2 −2gh_1 ))) =((160±(√(160^2 −2×32.2×350)))/(32.2)) =6.687 s or 3.251 s t=3.687 s or 0.251 s v_2 t−((gt^2 )/2)=h_2 =350 ⇒v_2 =((350+((32.2×3.687^2 )/2))/(3.687))=154.289 ft/s or ⇒v_2 =((350+((32.2×0.251^2 )/2))/(0.251))=1398.463 ft/s that means there are two possibilities. possibility 1: ball 2 with initial velocity 1398.463 ft/s or possibility 2: ball 2 with initial velocity 154.289 ft/s. (2) in possibility 1 the ball 1 is ascending and in possibility 2 the ball 1 is descending. (3) ball 1: v_1 t−((gt^2 )/2)=0 t=((2v_1 )/g)=((2×160)/(32.2))=9.938 s ball 2: t=((2v_2 )/g)=((2×1398.463)/(32.2))=86.861 s (possibilty 1) t=((2v_2 )/g)=((2×154.289)/(32.2))=9.583 s (possibilty 1)

$${g}=\mathrm{32}.\mathrm{2}\:{ft}/{s}^{\mathrm{2}} \\ $$$$\left(\mathrm{1}\right) \\ $$$${h}_{\mathrm{1}} ={v}_{\mathrm{1}} \left({t}+\mathrm{3}\right)−\frac{{g}\left({t}+\mathrm{3}\right)^{\mathrm{2}} }{\mathrm{2}} \\ $$$${h}_{\mathrm{2}} ={v}_{\mathrm{2}} {t}−\frac{{gt}^{\mathrm{2}} }{\mathrm{2}} \\ $$$${h}_{\mathrm{1}} ={h}_{\mathrm{2}} =\mathrm{350}\:{ft} \\ $$$${v}_{\mathrm{1}} \left({t}+\mathrm{3}\right)−\frac{{g}\left({t}+\mathrm{3}\right)^{\mathrm{2}} }{\mathrm{2}}={h}_{\mathrm{1}} =\mathrm{350} \\ $$$${t}+\mathrm{3}=\frac{\mathrm{1}}{{g}}\left({v}_{\mathrm{1}} \pm\sqrt{{v}_{\mathrm{1}} ^{\mathrm{2}} −\mathrm{2}{gh}_{\mathrm{1}} }\right) \\ $$$$\:\:\:\:\:=\frac{\mathrm{160}\pm\sqrt{\mathrm{160}^{\mathrm{2}} −\mathrm{2}×\mathrm{32}.\mathrm{2}×\mathrm{350}}}{\mathrm{32}.\mathrm{2}} \\ $$$$\:\:\:\:=\mathrm{6}.\mathrm{687}\:{s}\:{or}\:\mathrm{3}.\mathrm{251}\:{s} \\ $$$${t}=\mathrm{3}.\mathrm{687}\:{s}\:{or}\:\mathrm{0}.\mathrm{251}\:{s} \\ $$$${v}_{\mathrm{2}} {t}−\frac{{gt}^{\mathrm{2}} }{\mathrm{2}}={h}_{\mathrm{2}} =\mathrm{350} \\ $$$$\Rightarrow{v}_{\mathrm{2}} =\frac{\mathrm{350}+\frac{\mathrm{32}.\mathrm{2}×\mathrm{3}.\mathrm{687}^{\mathrm{2}} }{\mathrm{2}}}{\mathrm{3}.\mathrm{687}}=\mathrm{154}.\mathrm{289}\:{ft}/{s} \\ $$$${or} \\ $$$$\Rightarrow{v}_{\mathrm{2}} =\frac{\mathrm{350}+\frac{\mathrm{32}.\mathrm{2}×\mathrm{0}.\mathrm{251}^{\mathrm{2}} }{\mathrm{2}}}{\mathrm{0}.\mathrm{251}}=\mathrm{1398}.\mathrm{463}\:{ft}/{s} \\ $$$${that}\:{means}\:{there}\:{are}\:{two}\:{possibilities}. \\ $$$${possibility}\:\mathrm{1}:\:{ball}\:\mathrm{2}\:{with}\:{initial} \\ $$$${velocity}\:\mathrm{1398}.\mathrm{463}\:{ft}/{s}\:{or} \\ $$$${possibility}\:\mathrm{2}:\:{ball}\:\mathrm{2}\:{with}\:{initial} \\ $$$${velocity}\:\mathrm{154}.\mathrm{289}\:{ft}/{s}. \\ $$$$\left(\mathrm{2}\right) \\ $$$${in}\:{possibility}\:\mathrm{1}\:{the}\:{ball}\:\mathrm{1}\:{is}\:{ascending} \\ $$$${and}\:{in}\:{possibility}\:\mathrm{2}\:{the}\:{ball}\:\mathrm{1}\:{is}\: \\ $$$${descending}. \\ $$$$\left(\mathrm{3}\right) \\ $$$${ball}\:\mathrm{1}: \\ $$$${v}_{\mathrm{1}} {t}−\frac{{gt}^{\mathrm{2}} }{\mathrm{2}}=\mathrm{0} \\ $$$${t}=\frac{\mathrm{2}{v}_{\mathrm{1}} }{{g}}=\frac{\mathrm{2}×\mathrm{160}}{\mathrm{32}.\mathrm{2}}=\mathrm{9}.\mathrm{938}\:{s} \\ $$$${ball}\:\mathrm{2}: \\ $$$${t}=\frac{\mathrm{2}{v}_{\mathrm{2}} }{{g}}=\frac{\mathrm{2}×\mathrm{1398}.\mathrm{463}}{\mathrm{32}.\mathrm{2}}=\mathrm{86}.\mathrm{861}\:{s}\:\left({possibilty}\:\mathrm{1}\right) \\ $$$${t}=\frac{\mathrm{2}{v}_{\mathrm{2}} }{{g}}=\frac{\mathrm{2}×\mathrm{154}.\mathrm{289}}{\mathrm{32}.\mathrm{2}}=\mathrm{9}.\mathrm{583}\:{s}\:\left({possibilty}\:\mathrm{1}\right) \\ $$

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