PROBLEMS gALVZIRGMULTIPLE-CONCEPT Example 7 A Head-On Collision 7.11 illustrates

Question

PROBLEMS gALVZIRGMULTIPLE-CONCEPT Example 7 A Head-On Collision 7.11 illustrates an elastic head-on collision between two balls. One ball has a -ess of m1 = 0.250 kg and an initial velocity of +5.00 m/s. The other has a mass of Before collision 0.800 kg and is initially at rest. No external forces act on the balls. What are the velocities of the balls after the collision? ReasoningT so that the kinetic energy of the two-ball system is the same before and after the balls collide. The second fact is that the collision occurs head-on. This means that the veloci- ties before and after the balls collide all point along the same line. In other words, the collision occurs in one dimension. Lastly, no external forces act on the bails, with the After cllision systera is isolated and its total linear momentum is conserved. Figure 7.11 A 0.250-kg ball, traveling with We expect that ball 1. having the smaller mass, will rebound to the left after striking an initial velocity of vor ball 2, which is more massive. Ball 2 vill be driven to the right in the process. +5.00 m/s, undergoes an elastic collision with a 0.800-kg ball that is initially at rest. Knowns and Unknowns The data for this problem are given in the following table: Description Symbol Value Comment Explicit Data Mass of ball I Initial velocity of ball 1 Mass of ball 2 0.250 kg +5.00 m/s 0.800 kg Before collision occurs. Implicit Data Initial velocity of ball 2 0 m/s Before collision occurs, ball is initially at rest. Unknown Variables Final velocity of ball 1 Final velocity of ball 2 After collision occurs. After collision occurs. Modeling the Problem

Explanation / Answer

let
m1 = 0.25 kg, u1 = 5 m/s

m2 = 0.8 kg, u2 = 0

v1 = ? and v2 = ?

Apply conservation of momentum

m1*u1 = m1*v1 + m2*v2   ----(1)

m1*(u1 - v1) = m2*v2 -----(2)

Apply conservation of kinetic energy

(1/2)*m1*u1^2 = (1/2)m1*v1^2 + (1/2)*m2*v2^2

m1*u1^2 = m1*v1^2 + m2*v2^2

m1*(u1^2 - v1^2) = m2*v2^2

m1*(u1 + v1)*(u1 - v1) = m2*v2^2 -----(3)

take equation (3)/equation(2)

u1+v1 = v2 ----(4)

sunstitute above relation in equation(1)

m1*u1 = m1*v1 + m2*(u1+v1)

m1*u1 - m2*u1 = m1*v1 + m2*v1

(m1 - m2)*u1 = v1*(m1+m2)

==> v1 = (m1-m2)*u1/(m1+m2)

= (0.25 - 0.8)*5/(0.25 + 0.8)

= -2.62 m/s <<<<<<<<<<<---------------------Answer

From equation(4) v1 = v2 - u1

substitute above value in equation 1

m1*u1 = m1*(v2 - u1) + m2*v2

m1*u1 + m1*u1 = m1*v2 + m2*v2

2*m1*u1 = v2*(m1+m2)

==> v2 = 2*m1*u/(m1+m2)

= 2*0.25*5/(0.25 + 0.8)

= 2.38 m/s <<<<<<<<<<<<<<<<<<----------------Answer

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