A ball with a mass of #3 kg# is rolling at #3 m/s# and elastically collides with a resting ball with a mass of #1 kg#. What are the post-collision velocities of the balls?

Answer 1

Equations of conservation of energy and momentum.

#u_1'=1.5m/s#

#u_2'=4.5m/s#

As wikipedia suggests:

#u_1'=(m_1-m_2)/(m_1+m_2)*u_1+(2m_2)/(m_1+m_2)*u_2=#
#=(3-1)/(3+1)*3+(2*1)/(3+1)*0=#
#=2/4*3=1.5m/s#
#u_2'=(m_2-m_1)/(m_1+m_2)*u_2+(2m_1)/(m_1+m_2)*u_1=#
#=(1-3)/(3+1)*0+(2*3)/(3+1)*3=#
#=-2/4*0+6/4*3=4.5m/s#

[Equations' source]

Derivation

Conservation of momentum and energy state:

Momentum

#P_1+P_2=P_1'+P_2'#
Since momentum is equal to #P=m*u#
#m_1*u_1+m_2*u_2=m_1*u_1'+m_2*u_2'# - - - #(1)#

Energy

#E_1+E_2=E_1'+E_2'#
Since kinetic energy is equal to #E=1/2*m*u^2#
#1/2*m_1*u_1^2+1/2*m_2*u_2^2=1/2*m_1*u_1^2'+1/2*m_2*u_2^2'# - - - #(2)#
You can use #(1)# and #(2)# to prove the equations mentioned above. (I tried but kept getting two solutions, which is not right)
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Answer 2

To find the post-collision velocities of the balls, we can use the principle of conservation of momentum and kinetic energy. The equation for the conservation of momentum is:

(m1v1_initial + m2v2_initial = m1v1_final + m2v2_final)

Using the given values:

(3 kg * 3 m/s + 1 kg * 0 m/s = 3 kg * v1_final + 1 kg * v2_final)

Solving for v1_final and v2_final, we find:

(9 kg*m/s = 3 kg * v1_final + 1 kg * v2_final)

Since the collision is elastic, kinetic energy is conserved as well:

(0.5 * m1 * v1_initial^2 + 0.5 * m2 * v2_initial^2 = 0.5 * m1 * v1_final^2 + 0.5 * m2 * v2_final^2)

Plugging in the given values:

(0.5 * 3 kg * (3 m/s)^2 + 0.5 * 1 kg * (0 m/s)^2 = 0.5 * 3 kg * v1_final^2 + 0.5 * 1 kg * v2_final^2)

Solving for v1_final and v2_final, we find:

(13.5 J = 4.5 J + 0.5 * 3 kg * v1_final^2 + 0.5 * 1 kg * v2_final^2)

(9 J = 0.5 * 3 kg * v1_final^2 + 0.5 * 1 kg * v2_final^2)

Using the momentum equation, we found (v1_final = 0 m/s) and (v2_final = 9 m/s).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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