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Question
Prove mathematically F = ma
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Solution
Derivation of F = ma from Newton’s Second Law of Motion:
Newton introduced the concept of momentum and say “The momentum of a moving body is defined as the product of its mass and velocity.”
Thus, p = mv, where p = momentum of body
m = mass of body
v = velocity of body
Suppose the velocity of body of mass m changes from u to v in time t.
Initial momentum, p1 = mu
and final momentum, p2 = mv
the change in momentum, (p2 − p1) takes place in time t. Then according to Newton’s second law of motion, the magnitude of force F is:
`("P"_2-"P"_1)/"t"` α F, or F = `("km"("v"-"u"))/"t"`
Where k = constant of proportionality
Now, `"a"(("v"-"u"))/"t"`, where a = acceleration of body
∴ F = k ma
⇒ F = ma .............(∵ k = 1, constant)
This relation holds good when mass of the body remains constant.
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