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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

  1. They meet at time t equals fraction numerator u over denominator g end fraction and at a height fraction numerator u to the power of 2 end exponent over denominator 2 G end fraction plus fraction numerator g T to the power of 2 end exponent over denominator 8 end fraction  
  2. 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 plus fraction numerator g T to the power of 2 end exponent over denominator 8 end fraction  
  3. 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  
  4. 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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