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Step-by-Step Solution
Step 1: Identify the Known Quantities
Let the total mass of the original block before splitting be M and its initial speed be 40\,\text{m s}^{-1} . After splitting into two equal parts, each part has mass \frac{M}{2} . One part is given to move at 60\,\text{m s}^{-1} in the same direction; let the other part move with an unknown speed v_2 .
Step 2: Apply Conservation of Momentum
The initial momentum of the block is:
P_{\text{initial}} = M \times 40.
After splitting, the total momentum must remain the same. Hence,
\left(\frac{M}{2}\right)\times 60 + \left(\frac{M}{2}\right)\times v_2 = M \times 40.
Simplify to find v_2 :
\frac{M}{2} \times 60 + \frac{M}{2} \times v_2 = 40M
30M + \frac{M}{2}v_2 = 40M
\frac{M}{2} v_2 = 40M - 30M = 10M
v_2 = \frac{10M \times 2}{M} = 20\,\text{m s}^{-1}.
Step 3: Calculate the Initial Kinetic Energy
The initial kinetic energy is
K_{\text{initial}} = \frac{1}{2} M \times (40)^2 = \frac{1}{2} M \times 1600 = 800M.
Step 4: Calculate the Final Kinetic Energy
After splitting, the kinetic energy is the sum of the kinetic energies of the two parts:
K_{\text{final}}
= \frac{1}{2} \left(\frac{M}{2}\right)(60)^2
+ \frac{1}{2} \left(\frac{M}{2}\right)(20)^2.
Calculate each term:
\frac{1}{2} \left(\frac{M}{2}\right)(60)^2 = \frac{M}{4}\times 3600 = 900M,
\frac{1}{2} \left(\frac{M}{2}\right)(20)^2 = \frac{M}{4}\times 400 = 100M.
Therefore,
K_{\text{final}} = 900M + 100M = 1000M.
Step 5: Compute the Fractional Change in Kinetic Energy
The change in kinetic energy is
\Delta K = K_{\text{final}} - K_{\text{initial}} = 1000M - 800M = 200M.
The fractional change is
\frac{\Delta K}{K_{\text{initial}}}
= \frac{200M}{800M}
= \frac{1}{4}.
Thus, in the form x : 4 , we have x = 1 .
Step 6: Final Answer
The fractional change in kinetic energy is 1 : 4 , hence x = 1 .
