To determine The component of velocity and displacement and the terminal velocity of the bullet has to be determined. Explanation Write the expression for the acceleration due to drag on the bullet. a D = − k v Here, constant is k and velocity of the bullet is v . In x direction, acceleration is zero and in y direction the acceleration is equal to g . Write the expression for the acceleration in x direction a x = − k v x (I) Here, velocity component in x direction is v x . Write the expression for the acceleration in x direction. a y = − k v y − g (II) Here, velocity component in y direction is v y and acceleration due to gravity is g . Integrate the equation (I) with respect to time “ t ”. ∫ a x d t = − ∫ k v x d t ∫ d v x v x = − k ∫ d t ln ( v x ) = − k t + c (III) Apply boundary condition t = 0 and v x = v 0 cos θ . ln ( v x ) = − k t + c ln ( v 0 cos θ ) = ( − k × 0 ) + c c = ln ( v 0 cos θ ) Substitute ln ( v 0 cos θ ) for c in equation (III). ln ( v x ) = − k t + ln ( v 0 cos θ ) ln ( v x v 0 cos θ ) = − k t v x = e − k t v 0 cos θ (IV) Thus, the horizontal component of velocity is v x = e − k t v 0 cos θ . Similar integrate the equation (IV). ∫ v x d t = v 0 cos θ ∫ e − k t d t Use the relation d x = v x d t in the equation. ∫ d x = v 0 cos θ ∫ e − k t d t x = v 0 cos θ ( − e − k t k ) + c ′ (V) Here, constant is c ' and displacement is x . Thus, the horizontal component of displacement is x = v 0 cos θ ( − e − k t k ) + c ′ . Substitute 0 for x and 0 for t in equation (V) 0 = v 0 cos θ ( − e − k ( 0 ) k ) + c ′ c ' = v 0 cos θ k Substitute v 0 cos θ k for c ' in equation (V). x = v 0 cos θ ( − e − k t k ) + v 0 cos θ k x = v 0 cos θ k ( 1 − e − k t ) Similarly integrate the equation (II) to find out the vertical component of velocity. ∫ a y d t = − ∫ ( k v y + g ) d t Use the relation d v = a d t in the equation. ∫ d v y d t d t = − ∫ ( k v y + g ) d t ∫ d v y ( k v y + g ) = − k ∫ d t ln ( k v y + g ) = − k t + C (VI) Apply boundary equation in equation (VI). ln ( ( k × v 0 sin θ ) + g ) = ( − k × 0 ) + C C = ln ( k v 0 sin θ + g ) Substitute ln ( k v 0 sin θ + g ) for C in equation (VI)

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ISBN: 9781118885840

Author: James L. Meriam, L. G. Kraige, J. N. Bolton

Publisher: WILEY

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Chapter 2.4, Problem 96P

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The component of velocity and displacement and the terminal velocity of the bullet has to be determined.

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