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Units And Measurement

Question

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Three particles each of mass 20gm are moving with velocities $10 space straight i with hat on top space cm divided by straight s comma space space 12 space straight j with hat on top space cm divided by straight s space and space 20 space straight k with hat on top space cm divided by straight s$ respectively collide at the origin. After collision, one particle acquires the velocity $space space 6 space straight i with hat on top plus 3 space straight j with hat on top space$ and second particles acquires the velocity $negative 6 space straight i with hat on top space plus 8 space straight j with hat on top$ . Find the velocity of third fragment and velocity of the centre of mass of system after collision.

Solution

Here comma space we space have space Mass comma space straight m subscript 1 equals straight m subscript 2 equals straight m subscript 3 equals 20 gm straight u with rightwards harpoon with barb upwards on top subscript 1 equals 10 space straight i with hat on top comma straight u with rightwards harpoon with barb upwards on top subscript 2 equals 12 space straight j with hat on top comma space straight u with rightwards harpoon with barb upwards on top subscript 3 equals 20 space straight k with hat on top space straight v with rightwards harpoon with barb upwards on top subscript 1 equals 6 space straight i with hat on top minus 3 space straight j with hat on top comma space straight v with rightwards harpoon with barb upwards on top subscript 2 space equals negative 6 space straight i with hat on top plus 8 space straight j with hat on top comma straight v with rightwards harpoon with barb upwards on top subscript 3 space equals space ?

Now, using the law of conservation of momentum,

space straight m subscript 1 space straight u with rightwards harpoon with barb upwards on top subscript 1 plus straight m subscript 2 space straight u with rightwards harpoon with barb upwards on top subscript 2 plus straight m subscript 3 space straight u with rightwards harpoon with barb upwards on top subscript 3 equals straight m subscript 1 space straight v with rightwards harpoon with barb upwards on top subscript 1 plus straight m subscript 2 space straight v with rightwards harpoon with barb upwards on top subscript 2 plus straight m subscript 3 stack space straight v with rightwards harpoon with barb upwards on top subscript 3

space space stack space space straight v with rightwards harpoon with barb upwards on top subscript 3 space space equals space fraction numerator straight m subscript 1 space left parenthesis straight u with rightwards harpoon with barb upwards on top subscript 1 thin space minus straight v with rightwards harpoon with barb upwards on top subscript 3 right parenthesis space plus space straight m subscript 2 space left parenthesis straight u with rightwards harpoon with barb upwards on top subscript 2 minus straight v with rightwards harpoon with barb upwards on top subscript 2 right parenthesis plus straight m subscript 3 space straight u with rightwards harpoon with barb upwards on top subscript 3 over denominator straight m subscript 3 end fraction space space space space space space space space equals fraction numerator 20 space left curly bracket 10 space begin display style straight i with hat on top end style minus left parenthesis 6 space begin display style straight i with hat on top end style minus 3 space begin display style straight j with hat on top end style right parenthesis space right curly bracket space plus space 20 space left curly bracket 12 space begin display style straight j with hat on top end style minus left parenthesis negative 6 space begin display style straight i with hat on top end style plus 8 space begin display style straight j with hat on top end style right parenthesis space right curly bracket space plus space 20 cross times 20 space begin display style straight k with hat on top end style over denominator 20 end fraction space space space space space space space space space equals space 10 space straight i with hat on top plus 7 space straight j with hat on top plus 20 space straight k with hat on top

The velocity of centre of mass of system is,

stack straight v space with rightwards harpoon with barb upwards on top space equals space fraction numerator straight m subscript 1 space straight v with rightwards harpoon with barb upwards on top subscript 1 space plus straight m subscript 2 space straight v with rightwards harpoon with barb upwards on top subscript 2 plus straight m subscript 3 space straight v with rightwards harpoon with barb upwards on top subscript 3 over denominator straight m subscript 1 plus straight m subscript 2 plus straight m subscript 3 end fraction space space space space space equals space fraction numerator straight m subscript 1 space straight u with rightwards harpoon with barb upwards on top subscript 1 plus straight m subscript 2 space straight u with rightwards harpoon with barb upwards on top subscript 2 plus straight m subscript 3 space straight u with rightwards harpoon with barb upwards on top subscript 3 over denominator straight m subscript 1 plus straight m subscript 2 plus straight m subscript 3 end fraction space space space space space equals space fraction numerator 20 cross times 10 space begin display style straight i with hat on top end style plus 20 space straight x space 12 space begin display style straight j with hat on top end style plus 20 space straight x space 20 space begin display style straight k with hat on top end style over denominator 20 plus 20 plus 20 end fraction space space space space space equals space 10 over 3 straight i with hat on top plus 4 space straight j with hat on top plus 20 over 3 straight k with hat on top

Some More Questions From Units and Measurement Chapter

To measure a physical quantity X, the unit U is repeatedly used for n times. What is the magnitude of physical quantity?

Does the numerical factor in the quantitative measurement depend on the system of unit used?

Is one unit sufficient to measure a physical quantity?

Can a physical quantity have two or more units? Give example.

What are fundamental quantities?

How many fundamental quantities are there in mechanics?

How many fundamental quantities are there in Physics?

Name the fundamental quantities of mechanics.

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