Answer: -4m/s
Explanation:
This problem is a good example of Free Fall, where the main condition is that the initial velocity must be zero [tex]V_{o}=0[/tex], and the main equations for this situation as follows:
[tex]y=y_{o}+V_{o}t-\frac{1}{2}gt^{2}[/tex] Â (1)
[tex]V=V_{o}-gt[/tex] Â (2)
Where:
[tex]y=0[/tex] is the final height of the lunar lander
[tex]y_{o}=5m[/tex] Â is the initial height of the lunar lander (we are told the engine is cut off when the lander is 5.0m above the surface)
[tex]V_{o}=0[/tex] is the initial velocity of the lunar lander (we are told when the engine is off at 5m, the lander is in free fall)
[tex]t[/tex] is the time
[tex]g=1.6m/s^{2}[/tex] is the acceleration due to gravity on the moon
Having this clear, let's start finding [tex]t[/tex] from (1):
[tex]0=5m-\frac{1}{2}(1.6m/s^{2})t^{2}[/tex] Â (3)
[tex]t=\sqrt{\frac{2(5m)}{1.6m/s^{2}}}[/tex] Â (4)
[tex]t=2.5s[/tex] Â (5)
Substituting (5) in (2):
[tex]V=0-gt=-(1.6m/s^{2})(2.5s)[/tex] Â (6)
Finally:
[tex]V=-4m/s[/tex] Â Note the velocity is negative because of the downward trajectory of the lunar lander due to the free fall.
The speed of the lander is about 4.3 m/s
[tex]\texttt{ }[/tex]
Acceleration is rate of change of velocity.
[tex]\boxed {a = \frac{v - u}{t} }[/tex]
[tex]\boxed {d = \frac{v + u}{2}~t }[/tex]
[tex]\boxed {v^2 = u^2 + 2ad}[/tex]
where:
a = acceleration ( m/s² )
v = final velocity ( m/s )
u = initial velocity ( m/s )
t = time taken ( s )
d = distance ( m )
[tex]\texttt{ }[/tex]
Let us now tackle the problem!
[tex]\texttt{ }[/tex]
Given:
position of the lander above the surface = h = 5.0 m
initial downward speed of lander = u = 1.5 m/s
acceleration due to gravity on the moon = g = 1.6 m/s²
Asked:
final speed of the lander = v = ?
Solution:
[tex]v^2 = u^2 + 2gh[/tex]
[tex]v^2 = 1.5^2 + 2(1.6)(5.0)[/tex]
[tex]v^2 = 18.25[/tex]
[tex]v = \sqrt{18.25}[/tex]
[tex]v \approx 4.3 \texttt{ m/s}[/tex]
[tex]\texttt{ }[/tex]
The speed of the lander just before it touches the surfacet is about 4.3 m/s
[tex]\texttt{ }[/tex]
[tex]\texttt{ }[/tex]
Grade: High School
Subject: Physics
Chapter: Kinematics
