Explanation Given: The mass of the each disks A and B is m = 0.5 kg . The initial velocity of the disk A is ( v A ) 1 = 6 m/s . The initial velocity of the disk B is ( v B ) 1 = 4 m/s at the slope of 4 3 . The coefficient of the restitution is e = 0.75 . Draw the free body diagram of the disks as shown in Figure (1). Write the principle of conservation of linear momentum. ∑ m v 1 = ∑ m v 2 (I) Here, m is the mass, v is the velocity and the suffixes 1 , 2 indicates the initial and final stages. Refer Figure (1). Write the formula for coefficient of restitution ( e ) . e = ( v A ) 2 x − ( v B ) 2 x ( v B ) x − ( v A ) x (II) Here, v is the velocity, the suffixes A , B indicates the disks A and B and the suffixes x , y indicates the axes. Conclusion: According law conservation of linear momentum: Consider When the disk A collides with disk B . At Initial: Calculate the initial momentum. (When the disks moving opposite direction to each other, disk B is positive and disk A is negative) The initial momentum is ∑ m v 1 = m B ( v B ) 1 x + m A ( v A ) 1 x From Figure (1). The velocity of the disk B when resolved along x − a x i s is 4 × ( 3 5 ) m/s . Substitute 0.5 kg for m A , m B , − 6 m/s for ( v A ) 1 x and 4 × ( 3 5 ) m/s for ( v B ) 1 x . ∑ m v 1 = ( 0.5 kg ) × 4 × ( 3 5 ) m/s + ( 0.5 kg ) ( − 6 m/s ) = 1.2 − 3 = − 1.8 N ⋅ s At final: Calculate the final momentum. The final momentum is ∑ m v 2 = m A ( v A ) 2 x + m B ( v B ) 2 x Substitute 0.5 kg for m A , m B . ∑ m v 2 = 0.5 kg ( v A ) 2 x + 0.5 kg ( v B ) 2 x = 0.5 ( v A ) 2 x + 0.5 ( v B ) 2 x Calculate the velocity of the disks after collision. Substitute − 1.8 N ⋅ s for ∑ m v 1 and 0.5 ( v A ) 2 x + 0.5 ( v B ) 2 x for ∑ m v 2 in Equation (I). − 1.8 N ⋅ s = 0.5 ( v A ) 2 x + 0.5 ( v B ) 2 x 0.5 ( v A ) 2 x + 0.5 ( v B ) 2 x = − 1.8 (III) Calculate the coefficient of restitution ( e ) . Here, e = 0.75 . Substitute 0.85 for e , − 6 m/s for ( v A ) 1 and 4 × ( 3 5 ) m/s for ( v B ) 1 in Equation (II). 0.75 = ( v A ) 2 x − ( v B ) 2 x [ 4 × ( 3 5 ) m/s ] − ( − 6 m/s ) 0.75 = ( v A ) 2 x − ( v B ) 2 x 8.4 ( v A ) 2 x − ( v B ) 2 x = 0.75 ( 8.4 ) ( v A ) 2 x = 6.3 + ( v B ) 2 x (IV) Calculate the final velocity of the disks A and B

15th Edition

ISBN: 9780137569830

Author: HIBBELER

Publisher: VST

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Chapter 15, Problem 73P

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The velocity of each disks just after the collision and direction of the disk B .

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Chapter 15, Problem 72P

Chapter 15, Problem 74P

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The velocity of each disks just after the collision and direction of the disk B .

Solution Summary: The author illustrates the principle of conservation of linear momentum by drawing the free body diagram of the disks as shown in Figure (1).

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The velocity of each disks just after the collision and direction of the disk B .

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Chapter 15 Solutions