Explanation Calculation: Express the velocity of mass centre using conservation of momentum as follows: ( m A + m B ) v ¯ = m A v A + m B v B v ¯ = m A v A + m B v B m A + m B Express the velocities to the mass centre as follows: v ′ A = v A − v ¯ (1) Here, v ¯ is velocity of mass centre. Substitute m A v A + m B v B m A + m B for v ¯ in Equation (1). v ′ A = v A − m A v A + m B v B m A + m B = m B ( v A + v B ) m A + m B Express the velocities to the mass centre as follows: v ′ B = v B − v ¯ (2) Substitute m A v A + m B v B m A + m B for v ¯ in Equation (2). v ′ B = v B − m A v A + m B v B m A + m B = m A ( v A + v B ) m A + m B Express the energy ( E A ) using the relation as follows: E A = 1 2 m A v ′ A ⋅ v ′ A (3) Substitute m B ( v A + v B ) m A + m B for v ′ A in Equation (3). E A = 1 2 m A ( m B ( v A + v B ) m A + m B ) ⋅ ( m B ( v A + v B ) m A + m B ) = m A m B 2 ( v A + v B ) ( v A + v B ) 2 ( m A + m B ) 2 Express the energy ( E B ) using the relation as follows: E B = 1 2 m B v ′ B ⋅ v ′ B (4) Substitute m A ( v A + v B ) m A + m B for v ′ B in Equation (4). E B = 1 2 m B m A ( v A + v B ) m A + m B B ⋅ m A ( v A + v B ) m A + m B = m A 2 m B ( v A + v B ) ( v A + v B ) 2 ( m A + m B ) 2 Express the energy ( E A ) 0 , ( E B ) 0 test as follows: ( E A ) 0 = 1 2 m A v 0 2 ( E B ) 0 = 1 2 m B v 0 2 Write the expression for sever for automobile A as follows: S A = E A ( E A ) 0 (5) Here, E A is energy and ( E A ) 0 is energies from tests

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 14.2, Problem 14.36P

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Verify the given described collision between two automobile is (mAmB)2 times more severe for the automobile B than for automobile A.

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Chapter 14.2, Problem 14.35P

Chapter 14.2, Problem 14.37P

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4. A 750 kg hammer is used to drive a 2500 kg pile into some loosely packed sand. The hammer is released from a height of 2.5 m above the top of the pile and is seen to rebound to a maximum height of 0.15 m above the point of impact. Determine: (a) the velocity of the pile immediately after impact, (b) the coefficient of restitution, and (c) the average force exerted by the hammer on the pile if the impact takes place over 0.08 s. 2.5 m P Ans. vp, = -2.62 m/s, e = 0.62, Favg = 81.9 kN %3D

14.35 Two automobiles A and B, of mass mĄ and m³, respectively, are traveling in opposite directions when they collide head on. The impact is as- sumed perfectly plastic, and it is further assumed that the energy absorbed by each automobile is equal to its loss of kinetic energy with respect to a moving frame of reference attached to the mass center of the two-vehicle system. Denoting by EA and EB, respectively, the energy absorbed by auto- mobile A and by automobile B, (a) show that EA/EB = mB/mA, that is, the amount of energy absorbed by each vehicle is inversely proportional to its mass, (b) compute EA and EB, knowing that mA = 2400 kg and mg = 1350 kg and that the speeds of A and B are, respectively, 135 km/h and 90 km/h. VB B

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Chapter 14 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

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Verify the given described collision between two automobile is ( m A m B ) 2 times more severe for the automobile B than for automobile A .

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Verify the given described collision between two automobile is (mAmB)2 times more severe for the automobile B than for automobile A.

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Chapter 14 Solutions