MHT CET · Physics · Motion In Two Dimensions
A stone is projected vertically upwards with velocity ' \(\mathrm{V}\).
Another stone of same mass is projected at an angle of \(60^{\circ}\) with the vertical with the same speed (V). The ratio of their potential energies at the highest points of their journey, is
- A 1:1
- B 4:1
- C 3:2
- D 2:1
Answer & Solution
Correct Answer
(B) 4:1
Step-by-step Solution
Detailed explanation
Maximum height, \(\mathrm{h}=\frac{\mathrm{u}^2 \sin ^2 \theta}{2 \mathrm{~g}}\)
For the first stone \(\theta=90^{\circ}, \sin 90^{\circ}=1\)
\(\therefore \mathrm{h}_1=\frac{\mathrm{u}^2}{2 \mathrm{~g}}=\frac{\mathrm{V}^2}{2 \mathrm{~g}}\)
For the second stone, \(\mathrm{h}_2=\frac{\mathrm{v}^2 \sin ^2 30^{\circ}}{2 \mathrm{~g}}=\frac{\mathrm{V}^2}{8 \mathrm{~g}}\) The masses are same. Hence ratio of potential energies
\(\frac{\mathrm{U}_1}{\mathrm{U}_2}=\frac{\mathrm{h}_1}{\mathrm{~h}_2}=4\)
For the first stone \(\theta=90^{\circ}, \sin 90^{\circ}=1\)
\(\therefore \mathrm{h}_1=\frac{\mathrm{u}^2}{2 \mathrm{~g}}=\frac{\mathrm{V}^2}{2 \mathrm{~g}}\)
For the second stone, \(\mathrm{h}_2=\frac{\mathrm{v}^2 \sin ^2 30^{\circ}}{2 \mathrm{~g}}=\frac{\mathrm{V}^2}{8 \mathrm{~g}}\) The masses are same. Hence ratio of potential energies
\(\frac{\mathrm{U}_1}{\mathrm{U}_2}=\frac{\mathrm{h}_1}{\mathrm{~h}_2}=4\)
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