Q: A particle of mass 1 kg is moving with a variable speed, kt , where k = 1 m/s2. Find mean kinetic energy from t = 1 s to t = 2 s.
(a) 1 J
(b) 7/6 J
(c) 6/7 J
(d) 9/11 J
Ans: (b)
Mean K.E , $ \displaystyle K.E_m = \frac{\int_{1}^{2}K.E dt}{\int_{1}^{2}dt} $
$ \displaystyle K.E_m = \frac{\int_{1}^{2}\frac{1}{2}m v^2 dt}{\int_{1}^{2}dt} $
$ \displaystyle K.E_m = \frac{\int_{1}^{2}\frac{1}{2}m (kt)^2 dt}{\int_{1}^{2}dt} $
$ \displaystyle K.E_m = \frac{\frac{1}{2}m k^2\int_{1}^{2}t^2 dt}{\int_{1}^{2}dt} $