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Anonymous asked in Science & MathematicsPhysics · 2 months ago

# A 26.0 g object moving to the right at 23.0 cm/s overtakes and collides elastically with a 10.0 g ?

object moving in the same direction at 15.0 cm/s. Find the velocity of each object after the collision.

_____cm/s (26.0 g object)

_____cm/s (10.0 g object)

### 2 Answers

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• Favorite Answer

a variety of ways to do this, but always includes two rules for elastic collisions.

1) momentum is conserved

2) kinetic energy is conserved

To conserve momentum, the center of mass (CoM) does not change velocity due to collision. That center of mass is moving at

26.0(23.0) + 10.0(15.0) / (26.0 + 10.0) = 20.78 cm/s

To conserve kinetic energy, we can think of the CoM as an ideal spring returning to each mass that strikes it an identical speed in the opposite direction.

The CoM sees the 26.0 g object approach at

23.0 - 20.78 = 2.22 cm/s,

and will see it depart at -2.22 cm/s.

A ground based observer sees it depart at the velocity of the CoM plus the relative velocity

v = 20.78 - 2.22 = 18.55... ≈ 18.6 cm/s◄

The CoM sees the 10.0 g object approach at

15.0 - 20.78 = -5.78 cm/s

and will see it depart at 5.78 cm/s

A ground based observer sees it depart at the velocity of the CoM plus the relative velocity

v = 20.78 + 5.78 = 26.55... ≈ 26.6 cm/s◄

As both results are positive, they are both still moving to the right

Another way to do it would be to know that by conserving kinetic energy, the relative velocities of approach will equal the relative velocities of departure.

The relative velocity of approach is 23.0 - 15.0 = 8.0 cm/s

Writing the conservation of momentum equation becomes

26.0(23.0) + 10.0(15.0) = 26.0(v₁) + 10.0(v₂)

and we know that v₂ - v₁ = 8.0 so v₂ = v₁ + 8.0

26.0(23.0) + 10.0(15.0) = 26.0(v₁) + 10.0(v₁ + 8.0)

v₁ = 18.555... ≈ 18.6 cm/s◄

v₂ = 18.6 + 8.0 = 26.6 cm/s ◄

• Elastic collisions

v is velocity after the collision, u before

v₁ = (u₁(m₁–m₂) + 2m₂u₂) / (m₁ + m₂)

v₂ = (u₂(m₂–m₁) + 2m₁u₁) / (m₁ + m₂)

u₁ = 23

u₂ = 15

m₁ = 25

m₂ = 10

v₁ = (u₁(m₁–m₂) + 2m₂u₂) / (m₁ + m₂)v₂ = (u₂(m₂–m₁) + 2m₁u₁) / (m₁ + m₂)

v₁ = (23(25–10) + 2•10•15) / (25 + 10)v₂ = (15(10–25) + 2•25•23) / (25 + 10)

v₁ = (345 + 300) / (35)v₂ = (–225 + 1150) / (35)

v₁ = (645) / (35)v₂ = (925) / (35)

v₁ = 18.4 m/sv₂ = 26.4 m/s

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