Exploring the Depth: Solving a Kinematic Mystery in a Cave | 15m Deep

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In summary, the problem involves an explorer dropping a pebble into a cavern to determine its depth. The explorer hears the pebble hit the water after 1.8 seconds, and the speed of sound in the cave is 345 m/s. The question is asking for the depth of the cavern. To solve this problem, one needs to use kinematic equations and consider the distance the pebble falls and the distance the sound wave travels. The time it takes the pebble to reach the surface is also dependent on the drop height. After attempting various methods, the correct answer is found to be 15 meters.
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Homework Statement


An explorer drops a pebble into a cavern to see how deep it is. At 1.8 seconds, the explorer hears the pebble hit the water. The speed of sound travels at 345 m/s in the cave. How deep is the cavern?

This seems like a kinematic question but I can seem to get the right answer. Does anyone have any idea how to approach this? The answer is 15 meters btw.


Homework Equations



The kinematics equations.

The Attempt at a Solution



I've tried letting the time it takes the pebble to hit the ground to be 1.8 minus the time it takes for the sound to come back up. So I had something like t=1.8-d/Vs, where Vs is the speed of sound. I tried plugging that time into a kinematic formula but I had no luck.
 
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What do you know about the distance that the pebble falls and the distance that the sound wave travels? How does the time it takes the pebble to reach the surface depend on the drop height h?
 
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I also tried using the formula d=0.5gt^2, but that didn't give me the correct answer either.

I would approach this problem by first understanding the concept of kinematics and how it applies to this scenario. Kinematics deals with the motion of objects and can be described using equations that relate distance, time, and acceleration. In this case, we are dealing with the motion of the pebble as it falls and the sound as it travels back up to the explorer's ears.

To solve this problem, we can use the equation d = v*t, where d is the distance, v is the velocity, and t is the time. In this case, the distance we are trying to find is the depth of the cavern, and the velocity we are dealing with is the speed of sound.

We know that the time it takes for the sound to travel down to the water and back up to the explorer's ears is 1.8 seconds. Therefore, we can write the equation as:

d = (345 m/s) * 1.8 s

Solving for d, we get a depth of 621 m. However, this answer does not make sense because the depth of the cavern cannot be negative. This is where we need to consider the direction of motion.

In this scenario, the pebble is dropped from the top of the cavern and falls towards the bottom. The sound, on the other hand, travels down to the water and then back up to the explorer's ears. This means that the sound is traveling in the opposite direction of the pebble's motion. In kinematics, we use positive and negative signs to indicate the direction of motion. In this case, we can assign the downward direction as positive and the upward direction as negative.

Using this information, we can rewrite the equation as:

d = (-345 m/s) * 1.8 s

Solving for d, we get a depth of -621 m. This answer makes more sense as it indicates that the sound traveled a distance of 621 m in the opposite direction of the pebble's motion, which is towards the top of the cavern. However, this is still not the correct answer.

The key to solving this problem is to realize that the pebble and the sound do not start from the same point. The pebble is dropped from the top of the cavern, while the sound starts from the point where the pebble hits
 

What is a kinematic question?

A kinematic question is a type of scientific question that deals with the motion of objects, specifically their position, velocity, and acceleration.

What are the three main components of kinematics?

The three main components of kinematics are position, velocity, and acceleration. These quantities are used to describe the motion of an object.

What is the difference between speed and velocity?

Speed refers to the rate at which an object is moving, while velocity includes both the speed and direction of the object's motion.

How do you calculate acceleration?

Acceleration is calculated by dividing the change in velocity by the change in time. This can be represented by the equation a = (vf - vi)/t, where a is acceleration, vf is final velocity, vi is initial velocity, and t is time.

What are some real-world applications of kinematics?

Kinematics is used in fields such as physics, engineering, and sports to analyze and predict the motion of objects. It is also used in navigation and robotics to calculate the movement of vehicles and machines.

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