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Physics-

General

Easy

Question

A projectile is fired vertically upwards with an initial velocity $u$ . After an interval of T seconds a second projectile is fired vertically upwards, also with initial velocity is

They meet at time $t = \frac{u}{g}$ and at a height $\frac{u^{2}}{2 G} + \frac{g T^{2}}{8}$
They meet at time $t = \frac{u}{g} + \frac{T}{2}$ and at a height $\frac{u^{2}}{2 g} + \frac{g T^{2}}{8}$
They meet at time $t = \frac{u}{g} + \frac{T}{2}$ and at a height $\frac{u^{2}}{2 g} - \frac{g T^{2}}{8}$
They never meet

The correct answer is: They meet at time $t equals fraction numerator u over denominator g end fraction plus fraction numerator T over denominator 2 end fraction$ and at a height $fraction numerator u to the power of 2 end exponent over denominator 2 g end fraction minus fraction numerator g T to the power of 2 end exponent over denominator 8 end fraction$

For first projectile, $h subscript 1 end subscript equals u t minus fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent$
For second projectile, $h subscript 2 end subscript equals u open parentheses t minus T close parentheses minus fraction numerator 1 over denominator 2 end fraction g open parentheses t minus T close parentheses to the power of 2 end exponent$
When both meet i.e. $h subscript 1 end subscript equals h subscript 2 end subscript$
$u t minus fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent equals u open parentheses t minus T close parentheses minus fraction numerator 1 over denominator 2 end fraction g open parentheses t minus T close parentheses to the power of 2 end exponent$
$rightwards double arrow u T plus fraction numerator 1 over denominator 2 end fraction g T to the power of 2 end exponent equals g t T rightwards double arrow t equals fraction numerator u over denominator g end fraction plus fraction numerator T over denominator 2 end fraction$
and $h subscript 1 end subscript equals u open parentheses fraction numerator u over denominator g end fraction plus fraction numerator T over denominator 2 end fraction close parentheses minus fraction numerator 1 over denominator 2 end fraction g open parentheses fraction numerator u over denominator g end fraction plus fraction numerator T over denominator 2 end fraction close parentheses to the power of 2 end exponent$
$equals fraction numerator u to the power of 2 end exponent over denominator 2 g end fraction minus fraction numerator g T to the power of 2 end exponent over denominator 8 end fraction$

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