Question

Solve the following problem.

Derive the expression for power in terms of F, m, and t.

Answer 1

1. A constant force F is applied to a body of mass (m) initially at rest (u = 0).
2. We have, v = u + at
   ∴ v = 0 + at
   ∴ v = at        ....(1)
3. Now, power is the rate of doing work,
   ∴ P = `"dW"/"dt"`
   ∴ P = `"F"*"ds"/"dt"   ....[because "dW" = "F"*"ds"]`
4. But `"ds"/"dt" = "v"`, the instantaneous velocity of the particle.
   ∴ P = F·v       ....(2)
5. According to Newton’s second law,
   F = ma            .....(3)
6. Substituting equations (1) and (3) in equation (2)
   P = (ma)(at)
   ∴ P = ma²t
   ∴ P = `("m"^2"a"^2)/"m" xx "t"`
   ∴ P = `"F"^2/"m" "t"`
7. As F and m are constant, therefore, P ∝ t.