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A projectile can have the same range R for two angles of projection. If T1 and T2 be the time of flights in the two cases, then the product of the two time of flights is directly proportional to

  • 1/R2

  • 1/R

  • R

  • R2


C.

R

We know in advance that range of projectile is same for complementary angles i.e. for θ and (900 - θ )

straight T subscript 1 space equals space fraction numerator 2 straight u space sin space straight theta over denominator straight g end fraction

straight T subscript 2 space equals space fraction numerator 2 straight u space sin space left parenthesis 90 to the power of 0 minus straight theta right parenthesis over denominator straight g end fraction space equals space fraction numerator 2 space straight u space cos space straight theta over denominator straight g end fraction
and space straight R space equals space fraction numerator straight u squared space sin space 2 straight theta over denominator straight g end fraction
Therefore comma space straight T subscript 1 space straight T subscript 2 space equals space fraction numerator 2 space straight u space sin space straight theta over denominator straight g end fraction space straight x space fraction numerator 2 space straight u space cos space straight theta over denominator straight g end fraction
space equals space fraction numerator 2 straight u squared space left parenthesis 2 space sin space straight theta space cos space straight theta right parenthesis over denominator straight g squared end fraction
space equals space fraction numerator 2 straight u squared space left parenthesis sin space 2 straight theta right parenthesis over denominator straight g squared end fraction
space equals space 2 straight R divided by straight g
space equals space straight T subscript 1 straight T subscript 2 space proportional to space straight R

We know in advance that range of projectile is same for complementary angles i.e. for θ and (900 - θ )

straight T subscript 1 space equals space fraction numerator 2 straight u space sin space straight theta over denominator straight g end fraction

straight T subscript 2 space equals space fraction numerator 2 straight u space sin space left parenthesis 90 to the power of 0 minus straight theta right parenthesis over denominator straight g end fraction space equals space fraction numerator 2 space straight u space cos space straight theta over denominator straight g end fraction
and space straight R space equals space fraction numerator straight u squared space sin space 2 straight theta over denominator straight g end fraction
Therefore comma space straight T subscript 1 space straight T subscript 2 space equals space fraction numerator 2 space straight u space sin space straight theta over denominator straight g end fraction space straight x space fraction numerator 2 space straight u space cos space straight theta over denominator straight g end fraction
space equals space fraction numerator 2 straight u squared space left parenthesis 2 space sin space straight theta space cos space straight theta right parenthesis over denominator straight g squared end fraction
space equals space fraction numerator 2 straight u squared space left parenthesis sin space 2 straight theta right parenthesis over denominator straight g squared end fraction
space equals space 2 straight R divided by straight g
space equals space straight T subscript 1 straight T subscript 2 space proportional to space straight R

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Give three examples of vector quantities.

Force, impulse and momentum.
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Give three examples of scalar quantities.

Mass, temperature and energy
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What are the basic characteristics that a quantity must possess so that it may be a vector quantity?

A quantity must possess the direction and must follow the vector axioms. Any quantity that follows the vector axioms are classified as vectors. 


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What is a scalar quantity?

A physical quantity that requires only magnitude for its complete specification is called a scalar quantity.
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What is a vector quantity?

A physical quantity that requires direction along with magnitude, for its complete specification is called a vector quantity.
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