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Question

# Derive the relationship F = ma, where each symbol has its usual significance.

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Solution

## Suppose an object of mass, m is moving along a straight line with an initial velocity, u. It is uniformly accelerated to velocity, v in time, t by the application of a constant force, F throughout the time, t. The initial and final momentum of the object will be, p1 = mu and p2 = mv respectively. The change in momentum ∝ p2 – p1 ∝ mv – mu ∝ m × (v - u) The rate of change of momentum ∝ m(v−u)t The applied force, F ∝ m(v−u)t F = km(v−u)t F = kma Here, a = (v−u)t is the acceleration, which is the rate of change of velocity. The quantity, k is constant of proportionality. The SI units of mass and acceleration are kg and ms−2​ respectively. The unit of force is so chosen that the value of the constant, k becomes one. For this one unit of force is defined as the amount that produces an acceleration of 1 ms−2​​ in an object of 1 kg mass. That is, 1 unit of force = k × 1 kg × 1 ms−2​​ Thus, the value of k becomes 1. Hence, F = ma. The unit of force is kg ms−2​​ or newton, which has the symbol N. The second law of motion gives us a method to measure the force acting on an object as a product of its mass and acceleration.

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