Question

50PTS!!!!

A golf ball is hit at a velocity of 42.0 m/s at an angle of 42.2° on Earth and on the Moon where the acceleration due to gravity is 1.62 m/s². How much higher will the maximum height be on the Moon compared to the maximum height on Earth?

Answer 1

Approximately
`205\; {\rm m}` (assuming that
`g = 9.81\; {\rm m\cdot s^(-2)}` on Earth and that air resistance is negligible.)

Step-by-step explanation:

Initial vertical velocity of the golf ball:

`\begin{aligned} & (\text{initial vertical velocity}) \\ =\; & (\text{initial velocity})\, \sin(\text{angle of elevation}) \\ =\; & (42.0\; {\rm m\cdot s^(-1)}})\, \sin(42.2^(\circ)) \\ \approx\; & 28.212\; {\rm m\cdot s^(-1)}\end{aligned}`.

Let
`u_(y)` denote the initial vertical of the ball. When the ball is at maximum height, the vertical velocity
`v_(y)` of the ball will be
`0`. Let
`x_(y)` denote the vertical displacement of the ball (height of the ball.)

Let
`a_(y)` denote the vertical acceleration of the ball. Under the assumptions, the vertical acceleration of the ball during the flight will be constantly
`(-g)`.

The SUVAT equation
`v^(2) - u^(2) = 2\, a\, x` relates these quantities. Rearrange this equation to find the maximum vertical displacement of the ball (value of
`x_(y)` when
`v_(y) = 0\; {\rm m\cdot s^(-1)}`.)

`{v_(y)}^(2) - {u_(y)^(2) = 2\, a_(y)\, x_(y)`.

`\begin{aligned}x_(y) &= \frac{{v_(y)}^(2) - {u_(y)}^(2)}{2\, a_(y)} \end{aligned}`.

On the Earth,
`a_(y) = (-g) = (-9.81)\; {\rm m\cdot s^(-2)}`. Therefore:

`\begin{aligned}x_(y) &= \frac{{v_(y)}^(2) - {u_(y)}^(2)}{2\, a_(y)} \\ &\approx \frac{(0\; {\rm m\cdot s^(-1)})^(2) - (28.212\; {\rm m\cdot s^(-1)})^(2)}{2\, ((-9.81)\; {\rm m\cdot s^(-2)})} \\ &\approx 40.57\; {\rm m}\end{aligned}`.

On the Moon, it is given that
`g = 1.62\; {\rm m\cdot s^(-2)}`, such that
`a_(y) = (-g) = (-1.62)\; {\rm m\cdot s^(-2)}`. Therefore:

`\begin{aligned}x_(y) &= \frac{{v_(y)}^(2) - {u_(y)}^(2)}{2\, a_(y)} \\ &\approx \frac{(0\; {\rm m\cdot s^(-1)})^(2) - (28.212\; {\rm m\cdot s^(-1)})^(2)}{2\, ((-1.62)\; {\rm m\cdot s^(-2)})} \\ &\approx 245.65\; {\rm m}\end{aligned}`.

The difference between the maximum heights will be approximately:

`(245.65\; {\rm m}) - (40.57\; {\rm m}) \approx 205\; {\rm m}`.