Cliff Height Calculation: Physics Problem Solved

by ADMIN 49 views

Hey guys! Ever wondered how high a cliff climber is if they drop something? Let's dive into a cool physics problem that'll show you how to figure that out. We'll break down the steps nice and easy, so you'll be calculating heights like a pro in no time!

Understanding the Problem

So, picture this: A climber is chilling at the top of a cliff and accidentally drops their water bottle. The bottle isn't just falling, it's got a little oomph to it because the climber kind of nudged it downwards with an initial velocity of 2 m/s. Now, the bottle plummets for 15 seconds before SPLASH! It hits the ground. The big question we need to answer is: How high was the climber when this happened?

To tackle this, we're going to use our physics know-how, specifically the equations of motion that deal with constant acceleration. Remember, gravity is our constant accelerator here, pulling that water bottle down towards the earth. The key here is to dissect the problem, identify what we know, and figure out what formula will help us link those pieces of information to find the height.

Breaking Down the Givens

Before we even think about formulas, let's list out what we already know. This is like gathering our ingredients before we start cooking up a solution:

  • Initial velocity (v₀): The bottle's starting speed downwards is 2 m/s. It's important to note the direction here. Since we're dealing with downward motion, we'll consider it positive (makes the math easier down the road).
  • Time (t): The bottle falls for a whole 15 seconds. That's a pretty long drop!
  • Acceleration (a): This is where gravity comes in. The acceleration due to gravity is approximately 9.8 m/s². This means the bottle's speed increases by 9.8 meters per second every second it falls. We'll use the positive value since the direction is downwards.
  • Displacement (d): This is the cliff height which is what we want to find. We can represent it as 'd'.

Now that we've got our ingredients lined up, we can start thinking about which recipe (aka formula) to use.

Choosing the Right Formula

Okay, we've got our initial velocity, time, and acceleration, and we're hunting for the displacement (the cliff height). This screams for a specific equation of motion, one that ties all these variables together. There are a few equations of motion, but the one that fits perfectly here is:

d = v₀t + (1/2)at²

Let's break down what this formula means:

  • d is the displacement, the distance the object has traveled (what we're trying to find).
  • v₀t represents the distance the object would travel if it were moving at a constant speed equal to its initial velocity.
  • (1/2)at² represents the extra distance covered due to the acceleration. The longer the time, the greater the effect of acceleration.

This formula is awesome because it directly links our known values to our unknown, the cliff height. Now, it's just a matter of plugging in the numbers and crunching them out!

Calculating the Height

Alright, let's get to the fun part – plugging in those numbers and getting our answer! We've got our formula ready: d = v₀t + (1/2)at² and we've identified our variables:

  • v₀ = 2 m/s
  • t = 15 s
  • a = 9.8 m/s²

Time to substitute these values into the equation:

d = (2 m/s)(15 s) + (1/2)(9.8 m/s²)(15 s)²

Now, let's simplify this step-by-step:

  1. First part: (2 m/s)(15 s) = 30 meters. This is the distance the bottle would have traveled if it had just been falling at its initial speed, without gravity speeding it up.
  2. Second part: (1/2)(9.8 m/s²)(15 s)²
    • First, calculate 15² which equals 225.
    • Then, multiply 9.8 by 225, which gives us 2205.
    • Finally, take half of 2205, which equals 1102.5 meters. This is the extra distance the bottle covers because of gravity's pull.

Now, let's put it all together:

d = 30 meters + 1102.5 meters

d = 1132.5 meters

Whoa! That's a pretty tall cliff! So, the calculated height is 1132.5 meters. But wait a minute... let's take a step back and think about this answer in the context of the options given in the original problem. The options were A. 20 m, B. 30 m, C. 40 m, and D. 50 m. Our calculated answer is way off from these choices. This suggests there might be an error in the way the problem was initially presented or perhaps the options provided are incorrect. It’s always good to double-check your work and the given information to make sure everything aligns!

Why is My Answer Different from the Multiple Choices?

Okay, so we got a result of 1132.5 meters, which is significantly different from the multiple-choice options (20m, 30m, 40m, 50m). This discrepancy tells us something's not quite right, and it’s a super important lesson in problem-solving: always check if your answer makes sense in the context of the problem! Here's a breakdown of why our answer might be so different and what we should consider:

  1. Possible Errors in the Problem Statement:
    • Incorrect Time: 15 seconds is a long time for a falling object. If the height was only 20-50 meters, the time should be much shorter. Perhaps the time given was a typo.
    • Incorrect Initial Velocity: While 2 m/s is a reasonable initial downward velocity, it contributes relatively little to the overall distance compared to gravity acting over 15 seconds.
  2. Multiple Choice Options:
    • The multiple-choice answers might be incorrect. This happens sometimes in textbooks or quizzes. If all the options are significantly smaller than our calculated answer, it's a red flag.
  3. Our Calculations:
    • We should always double-check our calculations to make sure we haven't made a mistake in substituting values or simplifying the equation. However, in this case, our calculations seem correct based on the given information.

Re-evaluating the Problem and Estimating a More Realistic Time

Let’s imagine the cliff is actually one of the heights given in the multiple-choice answers, say around 40 meters (Option C). If that's the case, we can estimate how long it should take the bottle to fall and see if it makes more sense.

We can use the same equation, d = v₀t + (1/2)at², but this time we're solving for t (time) instead of d (distance). This makes the math a little trickier, as we'll end up with a quadratic equation. For simplicity's sake, let's approximate and think conceptually:

  • If there was no initial velocity: An object falls approximately 5 meters in the first second due to gravity (using the simplified formula d ≈ 5t²). To fall 40 meters, it would take a little less than 3 seconds (since 5 * 3² = 45).
  • With the initial velocity of 2 m/s: The bottle will fall slightly faster, so the time will be a bit less than 3 seconds.

This estimation suggests that if the cliff were around 40 meters high, the bottle would likely hit the ground in under 3 seconds, not 15 seconds. This further reinforces the idea that there's likely an issue with the original problem statement.

Conclusion

So, while we successfully used the physics equations to calculate a height of 1132.5 meters based on the information given, it's crucial to recognize that this answer doesn't align with the provided multiple-choice options or a realistic scenario. The most likely explanation is an error in the problem statement, specifically the time given (15 seconds). Always remember, guys, that in physics (and in life!), it's not just about crunching numbers, but also about thinking critically about your results and whether they make sense in the real world. If you encounter a problem like this, don't hesitate to question the givens and think about the broader context!