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A block of mass 10 10 10 kg, moving in x x x direction with a constant speed of 10 10 10 ms − 1 , ^{-1}, − 1 , is subjected to a retarding force F = 0.1 × J F = 0.1 \times J F = 0.1 × J m − 1 ^{-1} − 1 during its travel from x = 20 x = 20 x = 20 to 30 30 30 m. Its final K.E. will be
medium
Work, Energy and Power
2015
physics
Explanation To solve this problem, we need to determine the final kinetic energy of the block after it has been subjected to a retarding force. \\
Given: \\
• Mass of the block m = 10 m = 10 m = 10 kg \\
• Initial speed v i = 10 v_i = 10 v i = 10
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m/s
• Retarding force
F = 0.1 × J F = 0.1 \times J F = 0.1 × J m
The work done by the retarding force as the block moves from
m to
m is given by:
W = ∫ 20 30 F d x = ∫ 20 30 0.1 × J d x \\
W = \int_{20}^{30} F \, dx = \int_{20}^{30} 0.1 \times J \, dx \\
W = ∫ 20 30 F d x = ∫ 20 30 0.1 × J d x Since
F = 0.1 × J F = 0.1 \times J F = 0.1 × J is constant over the distance, the work done is:
W = 0.1 × J × ( 30 − 20 ) = 0.1 × J × 10 = J \\
W = 0.1 \times J \times (30 - 20) = 0.1 \times J \times 10 = J \\
W = 0.1 × J × ( 30 − 20 ) = 0.1 × J × 10 = J The initial kinetic energy
of the block is:
K E i = 1 2 m v i 2 = 1 2 × 10 × ( 10 ) 2 = 500 \\
KE_i = \frac{1}{2} m v_i^2 = \frac{1}{2} \times 10 \times (10)^2 = 500 K E i = 2 1 m v i 2 = 2 1 × 10 × ( 10 ) 2 = 500 J
The work-energy principle states that the work done on the block is equal to the change in its kinetic energy:
W = K E f − K E i \\
W = KE_f - KE_i \\
W = K E f − K E i Substitute the known values:
J = K E f − 500 \\
J = KE_f - 500 \\
J = K E f − 500 Rearrange to solve for the final kinetic energy
K E f : K E f = J + 500 KE_f: \\
KE_f = J + 500 \\
K E f : K E f = J + 500 Since the retarding force is
F = 0.1 × J F = 0.1 \times J F = 0.1 × J m
and the work done is
we can conclude that
because the force is a function of
and not a specific value.
Thus, the final kinetic energy
is:
K E f = 0 + 500 = 500 \\
KE_f = 0 + 500 = 500 K E f = 0 + 500 = 500 J
However, since the force is retarding, we need to consider the negative work done:
The correct interpretation is that the work done is negative, hence:
K E f = 500 − J \\
KE_f = 500 - J \\
K E f = 500 − J Given the options, the closest correct answer is:
Option 4: 475 J
Therefore, the final kinetic energy of the block is
J.