Question

Derive the eqation f=ma

Answer 1

The law states that the rate of change of momentum of a body is proportonal to the Force 
producing it and takes place in the direction in which the force acts. 
Momentum of a body = P = mass* velocity = mV (Vector quantity) ; V = Velocity 
Let m = mass of the body 
Initial velocity = u ; initial momentum = mu (Vector) 
Final velocity = v ; final velocity = mv (vector) 
Time interval = t 
Change in velocity = m(v - u) (vector) 
Rate of change of momementum = m(v - u) / t = m{(v- u)/t} = ma (vector) ; a = acceleration (vector) 
Instantaneous rate of change of momentum = dP/dt = (d/dt)(mV) = m(dV/dt) = ma 
Let the force acting on the body be F (vector) 
According to the 2nd law : F proportional to ma 
=> F = k ma, (a vector equation), where k = constant ........ (1) 
Thus the directions of the Acceleration and the Force producing it are the same. 
Let us define Unit Force (Newton) as that force which when acts on a body of Unit Mass (1 kg) 
produces Unit Acceleration (1 m/s²). This means, Unit Force = k (Unit Mass)*(Unit Acceleration) 
=> k = 1 
Hence F = ma (a vector equation)

Answer 2

To prove: F=ma According to second law of motion, f is directly proportional to rate of change of momentum. F is directly proportional to mv-mu/ t F is directly proportional to m(v-u)/t F is directly proportional to ma F=kma(where k is constant and si unit of k is 1) F= ma Hence proved