Two projectiles of same mass have their maximum kinetic encrgies in ratio 4: 1 and ratio of their maximum heights is also 4 : 1 then what is the ratio of their ranges?

Options

A. `2:1`
B. `4:1`
C. `8:1`
D. `16:1`

Answer

B

Explanation

The kinetic energy of the projectiles is maximum at the point of release.
The ratio of K.E of projectiles
` ( ( 1 ) /( 2 ) m u _ 1 ^ 2 ) /( ( 1 ) /( 2 ) m u _ 2 ^ 2 ) = ( 4 ) / ( 1 ) `
` ( u _ 1 ^ 2 ) / ( u _ 2 ^ 2 ) = ( 4 ) /( 1 ) " " `... (i)
The ratio of maximum heights of projectiles
` ( H _ 1 ) /( H _ 2 ) = ( 4 ) / ( 1 ) `
` ( ( u _ 1 ^ 2 sin ^ 2 theta _ 1 ) /( 2 g ) ) `
`( u _ 1 ^ 2 )/ ( u _ 2 ^ 2 ) * ( sin ^ 2 theta _ 1 ) / ( sin ^ 2 theta _ 2 ) = ( 4 ) / ( 1 ) `
` ( 4 ) /( 1) *(sin ^ 2 theta _ 1 ) /( sin ^ 2 theta _ 2 ) = ( 4 ) / ( 1 )" "` ( from equation (i))
`sin ^ 2 theta _ 1 = sin ^ 2 theta _ 2 `
` theta _ 1 = theta _ 2 `
Now, ratio of ranges of projectiles
` (R _ 1 ) / ( R _ 2 ) = ( u _ 1 ^ 2 sin 2 theta _ 1 ) / ( u _ 2 ^ 2 sin 2 theta _2 ) `
` ( R _ 1 ) / ( R _ 2 ) = ( u _ 1 ^ 2 sin 2 theta _ 1 ) /( u _ 2 ^ 2 sin 2 theta _ 1 ) " " ( because theta _ 1 = theta _ 2 ) `
` ( R _ 1 ) / ( R _2 ) = ( u _ 1^ 2 ) /( u _ 2^ 2 ) , ( R _ 1 ) /( R _ 2 ) = ( 4 ) /( 1 ) `