A) After time (they all lie on a parabola
B) After time (they all lie on a circle
C) After time t parabola described by particles has focal distance ut
D) After time t circle described by particles has radius 2ut
Correct Answer: B
Solution :
| [b] For any projected particle, | ||
| \[x=u\cos \theta \cdot t\] | ... (i) | |
| \[y=u\sin \theta \cdot t-\frac{1}{2}g{{t}^{2}}\] | ... (ii) | |
| Squaring both Eqs. (i) and (ii), we get | ||
| \[{{x}^{2}}={{u}^{2}}{{\cos }^{2}}\theta \cdot {{t}^{2}}\] | ... (iii) | |
| \[{{y}^{2}}={{u}^{2}}{{\sin }^{2}}\theta \cdot {{t}^{2}}-\left( \frac{1}{2}g{{t}^{2}} \right)\] | ... (iv) | |
| Now, adding Eqs. (iii) and (iv), we get |
| \[{{u}^{2}}({{\cos }^{2}}\theta +{{\sin }^{2}}\theta ){{t}^{2}}={{x}^{2}}+{{\left( y+\frac{1}{2}g{{t}^{2}} \right)}^{2}}\] |
| \[\Rightarrow \] \[{{u}^{2}}{{t}^{2}}={{x}^{2}}+{{\left( y+\frac{1}{2}gt \right)}^{2}}\] |
| This is a circle with centre \[\left( 0,-\frac{1}{2}g{{t}^{2}} \right)\]and radius ut. |
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