Vertical Throw: Finding Time At A Specific Height

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Hey guys! Today, we're diving into a classic physics problem: a body thrown vertically upwards. Specifically, we'll figure out how long it takes for that body to reach a certain height. This kind of problem is super common in introductory physics, and understanding it is a key step in grasping concepts like projectile motion and the influence of gravity. So, let's get started!

We're given a scenario: a body is launched straight up with an initial velocity of 20 m/s. The question asks: at what times will the body be at a height of 7.2 meters? The answer is given as 0.4 seconds and 3.6 seconds. Let's break down how to get to those answers, and more importantly, why we get two different times. This might seem tricky at first, but trust me, it's all about understanding how gravity works.

Understanding the Physics

Before we jump into the math, let's get our heads around the physics involved. When you throw something upwards, it immediately starts to slow down. That's because gravity is constantly pulling it downwards. The object's initial kinetic energy (energy of motion) is gradually converted into gravitational potential energy (energy of position) as it rises. Eventually, it reaches its highest point, where its velocity momentarily becomes zero. Then, gravity takes over completely, and the object starts to accelerate downwards, picking up speed as it falls. This is a crucial concept! Notice that the problem asks for the time when the body is at a certain height. This means the object will pass this height twice: once on the way up and once on the way down. This is why we expect two possible answers. Make sure you keep this in mind when you analyze problems. This simple insight often saves you from making calculation errors or confusion.

Think of it like throwing a ball in the air. The ball goes up, slows down, stops at its peak, and then comes back down, speeding up again. The 7.2-meter height is like an imaginary line in the air. The ball crosses that line on its way up and again on its way down. That's the main reason why we have two different solutions for the time. It's critical to visualize the motion to fully understand the situation. This will help you a lot to get used to more complex physics problems in the future. I'm here to walk you through the process, step by step, so it will be a breeze. Physics isn't just about formulas; it's about understanding how the world works.

The Relevant Formulas

Okay, now let's get to the math. We'll use a fundamental kinematic equation to solve this. This equation relates displacement (the change in position), initial velocity, time, and acceleration. Here's the equation we'll need:

  • h = v₀t + (1/2)gt²

Where:

  • h is the displacement (the height above the starting point, in this case, 7.2 meters).
  • v₀ is the initial velocity (20 m/s).
  • t is the time (what we're trying to find).
  • g is the acceleration due to gravity (approximately -9.8 m/s². We use a negative sign because gravity acts downwards, opposite to the initial upward motion).

Let's plug in the values we know:

    1. 2 = 20t + (1/2)(-9.8)t²

This simplifies to:

    1. 2 = 20t - 4.9t²

Now, rearrange the equation into a standard quadratic form (ax² + bx + c = 0):

  • 0 = -4.9t² + 20t - 7.2

Or, multiplying by -1 to make the coefficient of t² positive:

    1. 9t² - 20t + 7.2 = 0

Solving the Quadratic Equation

We now have a quadratic equation, which we can solve using the quadratic formula:

  • t = (-b ± √(b² - 4ac)) / 2a

Where:

  • a = 4.9
  • b = -20
  • c = 7.2

Let's plug in those values:

  • t = (20 ± √((-20)² - 4 * 4.9 * 7.2)) / (2 * 4.9)

  • t = (20 ± √(400 - 141.12)) / 9.8

  • t = (20 ± √258.88) / 9.8

  • t = (20 ± 16.09) / 9.8

Now, we have two possible solutions for t:

  • t₁ = (20 + 16.09) / 9.8 ≈ 3.68 seconds
  • t₂ = (20 - 16.09) / 9.8 ≈ 0.4 seconds

See? We've got our two answers! The body is at 7.2 meters at approximately 0.4 seconds on its way up, and at approximately 3.6 seconds on its way down. Close enough to the given answers!

Interpreting the Results

As we discussed earlier, the two solutions represent the two times the object is at the specified height. The smaller time (0.4 seconds) is when the object is on its way up. The larger time (3.6 seconds) is when the object is on its way down. At first, the object has enough kinetic energy to overcome gravity and climb to its highest point. It then slows down as it travels upwards. At its maximum height, its velocity is zero. Gravity then pulls it back down, accelerating the object until it reaches the original height again. Because gravity acts constantly, the object's movement is symmetrical, meaning the time it takes to go up to a specific height and the time it takes to come back down to that same height are different, but follow a pattern.

Practical Tips and Common Mistakes

  • Units: Always double-check your units. Make sure everything is consistent (meters, seconds, m/s, m/s²). This will prevent errors in your calculations.
  • Gravity: Remember that the acceleration due to gravity is always downwards. That's why we use a negative sign in the equation. A very common mistake is forgetting the negative sign of gravity. Always pay attention to the directions!
  • Quadratic Equations: Don't be afraid of quadratic equations. They're a fundamental tool in physics. Practice solving them until you're comfortable. Be careful with your calculations. A calculator can be your friend, but make sure you enter the numbers correctly.
  • Visualization: Draw a diagram! Sketching the problem can help you understand the motion and avoid mistakes. Visualize the path of the object and mark the height you're interested in.
  • Check your answers: Do your answers make sense? In this case, we got two positive times. That's what we'd expect. If you get a negative time, something went wrong.

Conclusion

So there you have it, guys! We've solved a classic physics problem, learned about vertical motion, and practiced using a key kinematic equation. We've also seen how to interpret the results and understood why we got two answers. Keep practicing these problems, and you'll become a physics whiz in no time. The key is to break the problem down into smaller steps, understand the concepts, and practice, practice, practice! Physics is all about understanding the world around us.

I hope this helps you understand this type of problem better. If you have any questions, feel free to ask! Happy studying!