ExampleA 45-kg swimmer runs with a horizontal velocity of +5.1 m/s off of a boat dock into a stationary 12-kg rubber raft. Find the velocity that the swimmer and raft would have after impact, if there were no friction and resistance due to the water.

Solution:

Given: m1 = 45 kg, m2 = 12 kg,

Find:

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o1

?v,v 21 ff

Consider motion of boy and raft just before and just after impact

Boy and raft define the system

Neglect friction and air resistance no external forces (which act in the direction of motion)

Therefore, we can use the Conservation of Linear Momentum

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2121

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Since, the boy moves with the raft after the impact

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1

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1.5)1245(

45v

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What if we have the case where vf1 vf2 ? We then have two unknowns. So, we need another equation.

This is the situation discussed in example 7

We can use Conservation of Mechanical Energy. No non-conservative forces. No change in y – so only KE. 2

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11212

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From original conservation of momentum equation, solve for vf2. Then substitute into conservation of energy equation.

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112

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)vv(vv

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mm

Eq. (7.8a)

Here is a trick!

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Eq. (7.8b)

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kg 230)1.5)(45(vkg 97)1.8)(12(v

kg 133)95.2)(45(v

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95.21.51245

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ff

ff

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Use numerical data from example

Momentum is conserved!

Collisions in 2D Start with Conservation of Linear Momentum vector equation

Similar to Newton’s 2nd Law problems, break into x- and y-components

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Example – Problem 7.34

Three guns are aimed at the center of a circle. They are mounted on the circle, 120° apart. They fire in a timed sequence, such that the three bullets collide at the center and mash into a stationary lump.

Two of the bullets have identical masses of 4.50 g each and speeds of v1 and v2. The third bullet has a mass of 2.50 g and a speed of 575 m/s. Find the unknown speeds.

Solution:

Given: m1 = m2 = 4.50 g, m3 = 2.50 g, vo3 = 575 m/s, vf1 = vf2 = vf3 = 0

Find: vo1 and vo2

Method: If we neglect air resistance then there are no external forces (in the horizontal x-y plane; gravity acts in the vertical direction) we can use Conservation of Linear Momentum

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