A robot probe drops a camera off the rim of a 330m high cliff on Mars, where the free-fall acceleration is 3.7m/s2. find the velocity with which it hits the ground.
The Correct Answer and Explanation is :
Given Data:
- Height of the cliff (( h )): ( 330 \, \text{m} )
- Acceleration due to gravity on Mars (( g )): ( 3.7 \, \text{m/s}^2 )
- Initial velocity (( u )): ( 0 \, \text{m/s} ) (as it is dropped from rest)
We can use the equation of motion:
[
v^2 = u^2 + 2gh
]
Since ( u = 0 ):
[
v^2 = 2gh
]
Taking the square root:
[
v = \sqrt{2gh}
]
Substituting the values:
[
v = \sqrt{2 \cdot 3.7 \cdot 330}
]
[
v = \sqrt{2442}
]
[
v \approx 49.42 \, \text{m/s}
]
Final Answer:
The velocity with which the camera hits the ground is approximately ( 49.42 \, \text{m/s} ).
Explanation (300 Words):
The scenario involves a camera dropped off a cliff on Mars, where the gravitational acceleration is different from Earth’s ( 9.8 \, \text{m/s}^2 ). Since the camera is dropped from rest, its initial velocity (( u )) is ( 0 ). The motion occurs under constant acceleration due to gravity, allowing us to use the kinematic equation:
[
v^2 = u^2 + 2gh
]
The equation relates the final velocity (( v )), initial velocity (( u )), gravitational acceleration (( g )), and height (( h )). Here, ( u^2 ) drops out because ( u = 0 ). The acceleration (( g )) and height (( h )) are known, so substituting these values simplifies the computation. The term ( 2gh ) calculates the square of the velocity the object gains after falling a distance ( h ) under acceleration ( g ).
The square root of ( 2gh ) gives the final velocity. On Mars, the lower gravity means the camera falls more slowly than it would on Earth. The value calculated, ( 49.42 \, \text{m/s} ), represents the speed at which the camera impacts the Martian ground.
This example illustrates the influence of planetary gravity on free-fall motion, which engineers and scientists consider when designing equipment for extraterrestrial exploration. The lower gravity on Mars results in gentler impacts for dropped objects compared to Earth.