An expression for the object's speed as it hits the ground is:
v = √ [ ( 2GMh ) / ( R ( R + h ) ) ]
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Further explanation
Let's recall the Gravitational Force formula:
[tex]\boxed {F = G\ \frac{m_1 m_2}{R^2}}[/tex]
where:
F = Gravitational Force ( N )
G = Gravitational Constant ( = 6.67 × 10⁻¹¹ Nm²/kg² )
m = mass of object ( kg )
R = distance between object ( m )
Let us now tackle the problem!
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Given:
mass of object = m
height position of object = h
mass of planet = M
radius of planet = R
initial speed of object = u = 0 m/s
Asked:
final speed of object = v = ?
Solution:
We will calculate the object's speed by using Conservation of Energy formula as follows:
[tex]Ep_1 + Ek_1 = Ep_2 + Ek_2[/tex]
[tex]-G \frac{Mm}{R + h} + \frac{1}{2}m u^2 = -G \frac{Mm}{R} + \frac{1}{2}m v^2[/tex]
[tex]-G \frac{Mm}{R + h} + \frac{1}{2}m (0)^2 = -G \frac{Mm}{R} + \frac{1}{2}m v^2[/tex]
[tex]-G \frac{Mm}{R + h} = -G \frac{Mm}{R} + \frac{1}{2}m v^2[/tex]
[tex]G \frac{Mm}{R} -G \frac{Mm}{R + h} = \frac{1}{2}m v^2[/tex]
[tex]G \frac{M}{R} -G \frac{M}{R + h} = \frac{1}{2} v^2[/tex]
[tex]v^2 = 2GM ( \frac{1}{R} -\frac{1}{R + h} )[/tex]
[tex]v^2 = 2GM\frac{h}{R(R +h) }[/tex]
[tex]\boxed {v = \sqrt { \frac{ 2GMh } { R(R +h) } } }[/tex]
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Learn more
- Unit of G : https://brainly.com/question/1724648
- Velocity of Runner : https://brainly.com/question/3813437
- Kinetic Energy : https://brainly.com/question/692781
- Acceleration : https://brainly.com/question/2283922
- The Speed of Car : https://brainly.com/question/568302
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Answer details
Grade: High School
Subject: Mathematics
Chapter: Gravitational Force
