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In a Broadway performance, an 80.0-kg actor swings from a 3.75-m-long cable that is horizontal when he starts. At the bottom of his arc, he picks up his 55.0-kg costar in an inelastic collision. What maximum height do they reach after their upward swing

Answer :

Explanation:

It is given that,

Mass of actor, m₁ = 80 kg

Length of cable, h₁ = 3.75 m

Mass of costar, m₂ = 55 kg

Using the conservation of energy to find the speed of the performer at the bottom of the arc as :

Potential energy at start = kinetic energy at end

[tex]mgh_1=\dfrac{1}{2}mv_1^2[/tex]

[tex]v_1=\sqrt{2gh_1}[/tex]

[tex]v_1=\sqrt{2\times 9.8\times 3.75}[/tex]

[tex]v_1=8.57\ m/s[/tex]

Let v₂ is its speed after the collision. Now using the conservation of momentum as :

[tex]m_1v_1=(m_1+m_2)v_2[/tex]

[tex]v_2=\dfrac{m_1v_1}{(m_1+m_2)}[/tex]

[tex]v_2=\dfrac{80\times 8.57}{80+55}[/tex]

[tex]v_2=5.07\ m/s[/tex]

Let h₂ is the maximum height they reach after their upward swing. Again using the conservation of energy as :

[tex]\dfrac{1}{2}mv_2^2=mgh_2[/tex]

[tex]h_2=\dfrac{v_2^2}{2g}[/tex]

[tex]h_2=\dfrac{(5.07)^2}{2\times 9.8}[/tex]

[tex]h_2=1.31\ m[/tex]

Hence, this is the required solution.

The maximum height which it has to reach after their upward swing is:

  • 1.31m

What is Maximum Height?

This refers to the highest vertical position which an object can reach starting from a rest point along an axis.

From the given set of data, we have

Mass of the actor, m₁ = 80 kg

Cable length, h₁ = 3.75 m

Mass of the costar, m₂ = 55 kg

We are using the conversion of energy:

Potential energy at start = kinetic energy at end

Therefore, v1 = 8.57 m/s

v2 = 5.07 m/s

Then to calculate for the maximum height, we would get:

1/2mv2= mgh2

h2= v²2/2g

h2= (5.07)²/2 x 9.8

h2= 1.31m.

This is the maximum height.

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