Let's say an object of mass 10 kg is fired from a cart of mass 90 kg. The object and the cart, of total mass 100 kg were initially moving together with a speed of 10 m/s. Then, the object is fired by some means from the cart such that it's velocity increases to 15 m/s (w.r.t ground), in the same direction. Finding the change in kinetic energy from ground and cart frame yields different values.
Calculating the kinetic energy difference in ground and cart frame of reference,
From Ground frame: $$m_1 = 10 kg, \ m_2 = 90 kg,\ u_1 = 10 m/s , \ u_2 = 10 m/s, v_1 = 15 m/s, v_2 = ?$$ Applying momentum conservation, $$ v_2 = 85/9 \ \ m/s$$ $$ \bigtriangleup KE = 1/2 ( \ 10 \times 15^2 + 90 \times (85/9)^2 - 100 \times 10^2 )$$ $$ = 1250/9 J \ = 138.888 J $$
From Cart frame (non-inertial): $$m_1 = 10 kg, \ m_2 = 90 kg,\ u_1 = 10 m/s , \ u_2 = 0 m/s, v_1 = (15 - 85/9) m/s = 50/9 m/s, v_2 = 0 m/s$$ $$ \bigtriangleup KE = 1/2 \times 10 \times (50/9)^2 $$ $$ = 154.321 J $$
There is a discrepancy of change in kinetic energy calculated from inertial and non-inertial frame. If one takes an inertial frame going with 10 m/s in the direction of the cart, the calculation from the frame will same change in kinetic energy as that given from the ground frame. What correction does the calculation from non-inertial frame need?
(Note: This problem deals with actual calculation of kinetic energy changes in a non-inertial frame. Here the pseudo-force is not constant/tractable, only impulse is.)
The question in K.E. with different frames is about calculating KE in different inertial frames. That question was resolved by saying that kinetic energy may be different for different inertial frames, but work energy theorem (related to change in kinetic energy) still holds. In this question,
- We have gone a step ahead and actually applied the work-energy theorem, rather than just calculate kinetic energy from different frames.
- The query here is about how to apply the work energy theorem for a non-inertial frame, to account for the change in kinetic energy.