Question #202911

A batter hits a baseball so that it leaves the bat with an initial velocity of 28.0m/s at an angle of 35°. What is the maximum height, the time it is in the air, and the range?

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1
Expert's answer
2021-06-06T21:36:01-0400

The information given is

  • The initial velocity is Vo=28.0  m/sV_{o}=28.0\;\text{m}/\text{s}
  • The launch angle is θ=350\theta=35^{0}
  • Gravity acceleration is ay=9.81  m/s2a_{y}=-9.81\;\text{m}/\text{s}^{2}
  • The starting position for the move is xo=0  m  and  yo=0  mx_{o}=0\;\text{m}\;\text{and}\;y_{o}=0\;\text{m}

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Components of the initial velocity.

Horizontal component.

cos(350)=VoxVocos(350)  Vo=V0xV0x=Vo  cos(350)Vox=28.0  m/s×cos(350)Vox=22.9  m/scos(35^{0})=\dfrac{V_{ox}}{V_{o}}\\ cos(35^{0})\;V_{o}=V_{0x}\\ V_{0x}=V_{o}\;cos(35^{0})\\ V_{ox}=28.0\;\text{m}/\text{s}\times cos(35^{0})\\ V_{ox}=22.9\;\text{m}/\text{s}


Vertical component.

sin(350)=VoyVosin(350)  Vo=V0yV0y=Vo  sin(350)Voy=28.0  m/s×sin(350)Voy=16.1  m/ssin(35^{0})=\dfrac{V_{oy}}{V_{o}}\\ sin(35^{0})\;V_{o}=V_{0y}\\ V_{0y}=V_{o}\;sin(35^{0})\\ V_{oy}=28.0\;\text{m}/\text{s}\times sin(35^{0})\\ V_{oy}=16.1\;\text{m}/\text{s}

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The maximum height is calculated with the following equation.

ΔYmax=Vfy2Voy22  ay\Delta Y_{max}=\dfrac{ V_{fy}^{2}-V_{oy}^{2}}{2\;a_{y}}

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Where.

VoyV_{oy} is the initial vertical velocity.

VfyV_{fy} is the velocity at the maximum height.

aya_{y} is the acceleration of gravity.

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Remember that at the top, the vertical component of velocity is zero.

Vfy=0  m/sV_{fy}=0\;\text{m}/\text{s}

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Evaluating numerically.

ΔYmax=Vfy2Voy22  ayΔYmax=(0  m/s)2(16.1  m/s)22×9.81  m/s2ΔYmax=13.2  m\Delta Y_{max}=\dfrac{ V_{fy}^{2}-V_{oy}^{2}}{2\;a_{y}}\\ \Delta Y_{max}=\dfrac{ (0\;\text{m}/\text{s} )^{2}-(16.1\;\text{m}/\text{s})^{2}}{2\times-9.81\;\text{m}/\text{s}^{2}}\\ \Delta Y_{max}=13.2\;\text{m}

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!!Answer.

The maximum height is ΔYmax=13.2  m\displaystyle \color{red}{\boxed{\Delta Y_{max}=13.2\;\text{m}}}

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Flight time is calculated using the following equation.

Vfy=Voy+ay  tV_{fy}=V_{oy}+a_{y}\;t

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where

VoyV_{oy} is the initial vertical velocity.

VfyV_{fy} is the final vertical velocity.

aya_{y} is the vertical acceleration.

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Remember that if the movement is symmetrical, the vertical velocity with which it reaches the ground is of the same magnitude but in the opposite direction

Vfy=VoyV_{fy}=-V_{oy} .

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Obtaining the expression for time.

Vfy=Voy+ay  tVoy+ay  t=Vfyay  t=VfyVoyt=VfyVoyayif  Vfy=Voyt=VoyVoyayt=2  VoyayV_{fy}=V_{oy}+a_{y}\;t\\ V_{oy}+a_{y}\;t=V_{fy}\\ a_{y}\;t=V_{fy}-V_{oy}\\ t=\dfrac{V_{fy}-V_{oy}}{a_{y} }\\ \text{if}\;V_{fy}=-V_{oy}\\ t=\dfrac{-V_{oy}-V_{oy}}{a_{y}}\\ t=\dfrac{-2\;V_{oy}}{a_{y}}

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Evaluating numerically.

t=2  Voyayt=2×16.1  m/s9.81  m/s2t=3.28  st=\dfrac{-2\;V_{oy}}{a_{y}}\\ t=\dfrac{-2\times 16.1\;\text{m}/\text{s}}{-9.81\;\text{m}/\text{s}^{2}}\\ t=3.28\;\text{s}

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!!Answer.

Flight time is t=3.28  s\displaystyle \color{red}{\boxed{t=3.28\;\text{s}}}

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The distance traveled is given by the following equation.

X=X0+Vox  tX=X_{0}+V_{ox}\;t

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where.

VoxV_{ox} is the horizontal velocity.

tt Is time.

XoX_{o} is the starting position.

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Evaluating numerically

X=X0+Vox  tX=0  m+22.9  m/s×3.28  sx=75.1  mX=X_{0}+V_{ox}\;t\\ X=0\;\text{m}+22.9\;\text{m}/\text{s}\times 3.28\;\text{s}\\ x=75.1\;\text{m}

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!!Answer

The maximum horizontal reach is x=75.1  m\displaystyle \color{red}{\boxed{x=75.1\;\text{m}}}


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