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Chapter 3: Problem 33

A hot-air balloon is ascending at the rate of \(12 \mathrm{~m} / \mathrm{s}\) and is \(80 \mathrm{~m}\) above the ground when a package is dropped over the side. (a) How long does the package take to reach the ground? (b) With what speed does it hit the ground?

Short Answer

Expert verified
(a) The package takes 5.74 s to reach the ground.(b) It hits the ground with a speed of 44.65 m/s.

Step by step solution

01

- Understand the Situation

A hot-air balloon is rising with a constant velocity of 12 m/s. When the package is dropped, it has an initial upward velocity of 12 m/s and is 80 m above the ground.
02

- Define Variables and Equations

Let the initial velocity be \(v_0 = 12 \mathrm{m/s}\), the height be \(h = 80 \mathrm{m}\), and the acceleration due to gravity be \(g = 9.8 \mathrm{m/s^2}\). Use the kinematic equation \[ h = v_0 t + \frac{1}{2}(-g)t^2 \] where we will solve for \(t\).
03

- Substitute Values and Rearrange Equation

Substitute the given values into the equation: \[ 0 = 80 + 12t - 4.9t^2 \]. Rearrange to form a standard quadratic equation: \[ 4.9t^2 - 12t - 80 = 0 \]
04

- Solve the Quadratic Equation

Use the quadratic formula \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 4.9\), \(b = -12\), and \(c = -80\). Calculate the discriminant first: \[ b^2 - 4ac = (-12)^2 - 4(4.9)(-80) = 144 + 1568 = 1712 \] Then, \[ t = \frac{12 \pm \sqrt{1712}}{9.8} \] gives the positive root \(t = 5.74 \mathrm{s} \)
05

- Find the Final Velocity

Use \(v = v_0 + (-g)t\) to find the final velocity just before hitting the ground. Substitute the values: \[ v = 12 - 9.8 \times 5.74 \approx -44.65 \mathrm{m/s} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Initial Velocity
Initial velocity, often denoted as \(v_0\), is the velocity of an object at the starting point of observation. In the context of kinematic equations, it's essential for determining how far and how fast something will move under uniform acceleration.

Here are some key points to understand:
  • When a package is dropped from a moving object, like a hot-air balloon, it retains the upward velocity of the balloon at the instant it is dropped. In this case, the initial velocity of the package is \(12 \text{ m/s} \) upwards.
  • This means that just at the moment of release, the package has the same upward motion as the balloon.
  • In problems involving kinematics, correctly identifying initial velocity helps to simplify and correctly apply the kinematic equations.
Understanding initial velocity sets the stage for calculating the object's future motion, allowing us to predict where and when an object will land.
Acceleration due to Gravity
The acceleration due to gravity, denoted as \(g\), is the rate at which an object's velocity changes due to the Earth's gravitational pull. It's roughly constant at \(9.8 \text{ m/s}^2\) downwards.

Here are some key concepts:
  • Gravity acts downward on all objects, pulling them towards Earth's center.
  • In the kinematic equation, gravity is usually considered as a negative acceleration when dealing with upwards initial velocity. Hence, \(a = -g = -9.8 \text{ m/s}^2\).
  • Acceleration due to gravity affects the motion of the package from the hot-air balloon, causing it to eventually descend until it hits the ground.
Even if the package starts with an upward velocity, gravity will slow it down, reverse its direction, and accelerate it back to the ground. This consistent force simplifies our calculations and helps us predict the object鈥檚 time of flight and impact velocity accurately.
Quadratic Equation
Quadratic equations appear frequently in kinematics problems, especially when dealing with objects under uniform acceleration, like gravity.

Key aspects of quadratic equations in physics:
  • They are formulated as \(at^2 + bt + c = 0\), where \(a\), \(b\), and \(c\) are constants.
  • In our problem, we used the kinematic equation \(h = v_0 t + \frac{1}{2}(-g)t^2\) and rearranged it to form a quadratic equation \(4.9t^2 - 12t - 80 = 0 \).
  • We solve quadratic equations using the quadratic formula: \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
For our exercise:
  • The coefficients were \(a = 4.9\), \(b = -12\), and \(c = -80\).
  • We calculated the discriminant: \(b^2 - 4ac = 1712\).
  • This gave us the time: \( t = \frac{12 \pm \sqrt{1712}}{9.8} \), resulting in a positive root of \( t = 5.74 \text{ s} \).
Solving the quadratic equation allowed us to find the time it took for the package to reach the ground, demonstrating the typical steps for such kinematic problems.

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Most popular questions from this chapter

A certain juggler usually tosses balls vertically to a height \(H\). To what height must they be tossed if they are to spend twice as much time in the air?

A parachutist bails out and freely falls \(50 \mathrm{~m}\). Then the parachute opens, and thereafter she slows at \(2.0 \mathrm{~m} / \mathrm{s}^{2}\). She reaches the ground with a speed of \(3.0 \mathrm{~m} / \mathrm{s}\). (a) How long is the parachutist in the air? (b) At what height does the fall begin?

Suppose you have the following equipment available: an electronic balance, a motion detector and an electronic force sensor attached to a computer-based laboratory system. You would like to determine the mass of a block of ice that can slide smoothly along a very level table top without noticeable friction. (a) Describe how you would use some of the equipment to find the gravitational mass of the ice. (b) Describe how you would use some of the equipment to find the inertial mass of the ice. (c) Which of the two types of masses can be measured in outer space where gravitational forces are very small?

A ball is thrown down vertically with an initial speed of \(v_{1}\) from a height of \(h .\) (a) What is its speed just before it strikes the ground? (b) How long does the ball take to reach the ground? What would be the answers to (c) part a and (d) part \(\mathrm{b}\) if the ball were thrown upward from the same height and with the same initial speed? Before solving any equations, decide whether the answers to (c) and (d) should be greater than, less than, or the same as in (a) and (b).

Imagine a landing craft approaching the surface of Callisto, one of Jupiter's moons. If the engine provides an upward force (thrust) of \(3260 \mathrm{~N}\), the craft descends at constant speed; if the engine provides only \(2200 \mathrm{~N}\), the craft accelerates downward at \(0.39\) \(\mathrm{m} / \mathrm{s}^{2}\). (a) What is the weight of the landing craft in the vicinity of Callisto's surface? (b) What is the mass of the craft? (c) What is the magnitude of the free-fall acceleration near the surface of Callisto?
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