Problem 54
Question
On our moon, the acceleration of gravity is \(1.6 \mathrm{m} / \mathrm{s}^{2} .\) If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 30 s later?
Step-by-Step Solution
Verified Answer
The rock will be traveling at 48 m/s just before it hits the bottom.
1Step 1: Identify the Given Values
We know the acceleration due to gravity on the moon is \( g = 1.6 \, \text{m/s}^2 \) and the time duration the rock falls is \( t = 30 \, \text{s} \).
2Step 2: Apply the Formula for Final Velocity
The formula to calculate the final velocity \( v \) of an object in free fall is \( v = g \times t \). Here, the rock is in free fall, with initial velocity \( u = 0 \).
3Step 3: Calculate the Final Velocity
Substitute the known values into the formula: \( v = 1.6 \, \text{m/s}^2 \times 30 \, \text{s} = 48 \, \text{m/s} \).
4Step 4: Conclusion
Thus, the rock's speed just before hitting the bottom of the crevasse is \( 48 \, \text{m/s} \).
Key Concepts
Acceleration Due to GravityFree FallFinal Velocity Calculation
Acceleration Due to Gravity
When we talk about acceleration due to gravity, we're referring to the force that pulls objects toward a massive body like a planet or a moon. This is a constant acceleration, meaning it acts on objects consistently and unstoppably.
The standard acceleration due to gravity on Earth is about 9.81 m/s². However, each celestial body has a different gravitational pull.
For example, on the Moon, the acceleration due to gravity is much weaker at 1.6 m/s². This lower gravity is why objects fall slower on the Moon compared to Earth.
This concept helps us understand how and why objects speed up when they are falling, as gravity is converting potential energy into kinetic energy as they descend.
The standard acceleration due to gravity on Earth is about 9.81 m/s². However, each celestial body has a different gravitational pull.
For example, on the Moon, the acceleration due to gravity is much weaker at 1.6 m/s². This lower gravity is why objects fall slower on the Moon compared to Earth.
This concept helps us understand how and why objects speed up when they are falling, as gravity is converting potential energy into kinetic energy as they descend.
- Gravity is a force that accelerates objects towards the mass center.
- Acceleration due to gravity varies by celestial body.
- Important for calculating how fast objects fall over time.
Free Fall
Free fall describes the motion of objects solely under the influence of gravity. This happens when an object is dropped, and no other forces—like air resistance—affect its descent.
In the vacuum of space, all objects, regardless of their mass, fall at the same rate. This is because acceleration due to gravity is constant for all masses.
On Earth, we're used to air resistance slowing things down, but on celestial bodies with no atmosphere, such as the Moon, objects in free fall continue to accelerate.
In the vacuum of space, all objects, regardless of their mass, fall at the same rate. This is because acceleration due to gravity is constant for all masses.
On Earth, we're used to air resistance slowing things down, but on celestial bodies with no atmosphere, such as the Moon, objects in free fall continue to accelerate.
- Free fall refers to motion under gravity alone.
- It allows for the acceleration calculation using gravity only.
- No air resistance or other forces present on bodies like the Moon.
Final Velocity Calculation
Calculating the final velocity of an object in free fall is straightforward when using the formula: \[ v = g \times t \] where \( v \) is the final velocity, \( g \) is the acceleration due to gravity, and \( t \) is the time the object has been falling.
This equation assumes the object started from rest, meaning its initial velocity \( u \) is zero.
For the Moon, using its gravitational acceleration of 1.6 m/s² and a fall time of 30 seconds, we calculate:
\[ v = 1.6 \, \text{m/s}^2 \times 30 \, \text{s} = 48 \, \text{m/s} \]
Thus, the object has a final velocity of 48 m/s. This straightforward computation shows how much speed an object can gain due to gravity alone over a specific period.
This equation assumes the object started from rest, meaning its initial velocity \( u \) is zero.
For the Moon, using its gravitational acceleration of 1.6 m/s² and a fall time of 30 seconds, we calculate:
\[ v = 1.6 \, \text{m/s}^2 \times 30 \, \text{s} = 48 \, \text{m/s} \]
Thus, the object has a final velocity of 48 m/s. This straightforward computation shows how much speed an object can gain due to gravity alone over a specific period.
- Use the formula \( v = g \times t \) for calculations.
- Assumes initial velocity is zero.
- Applies to environments with no air resistance, like on the Moon.
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