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Chapter 2: Problem 53

A student throws a water balloon vertically downward from the top of a building. The balloon leaves the thrower's hand with a speed of \(15.0 \mathrm{~m} / \mathrm{s}\). (a) What is its speed after falling freely for \(2.00 \mathrm{~s}\) ? (b) How far does it fall in \(2.00 \mathrm{~s} ?\) (c) What is the magnitude of its velocity after falling \(10.0 \mathrm{~m} ?\)

Short Answer

Expert verified
We'll use kinematic equations to find velocity and distance.

Step by step solution

01

Identify the Known Values

We have an initial velocity \(v_i = 15.0\, \mathrm{m/s}\), a time \(t = 2.00\, \mathrm{s}\), and acceleration due to gravity \(g = 9.81\, \mathrm{m/s^2}\).

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Key Concepts

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

Vertical Motion
Vertical motion refers to the movement of an object in the up or down direction along a single vertical axis. This type of motion can include both the initial force applied to the object and the influence of gravity as the object travels.
  • When we talk about a water balloon being thrown downward, it follows a vertical path as it moves towards the ground.
  • Even after the balloon is released, it continues to accelerate due to the influence of gravity.
  • Vertical motion can occur upwards, like jumping, or downwards, like falling.
Understanding vertical motion is vital in calculating how long it takes for an object to reach the ground and how fast it will be moving upon impact.
Initial Velocity
Initial velocity is the speed and direction an object has when it begins its motion. In problems involving free-fall, it's important to note whether the initial velocity is directed upwards or downwards. * In the exercise we have, the balloon is thrown downward with an initial velocity of 15.0 m/s. * This initial velocity is crucial because it determines the starting speed of the balloon as it falls. Knowing the initial velocity helps us calculate further specifics like how fast the object will be moving at any given time during its fall.
Acceleration Due to Gravity
Acceleration due to gravity is a constant force that affects all objects in free-fall near Earth's surface. * It's denoted by the symbol "g", and is approximately 9.81 m/s虏. * Gravity's acceleration does not change based on an object's size or weight. While the initial velocity gives the balloon its starting speed, gravity makes the balloon accelerate, increasing its speed as it moves downwards. This constant acceleration dictates the dynamics of free-falling objects, influencing calculations related to their speed and how far they鈥檝e traveled in a given period.
Displacement Calculation
Displacement refers to the overall change in position of the object. For vertical motion, displacement can be calculated using the equation:\[ s = v_i \times t + \frac{1}{2}gt^2 \]In this formula:
  • \( s \) represents displacement,
  • \( v_i \) is the initial velocity,
  • \( g \) is the acceleration due to gravity, and
  • \( t \) is the time in seconds.
By plugging in the known values:* Initial velocity (15.0 m/s)* Time (2.00 s)* Gravity (9.81 m/s虏)We can calculate how far the balloon fell during the time considered in the problem. Displacement calculation is critical in understanding how far and how fast an object has traveled from its original position.

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Most popular questions from this chapter

Nerve impulses travel at different speeds, depending on the type of fiber through which they move. The impulses for touch travel at \(76.2 \mathrm{~m} / \mathrm{s},\) while those registering pain move at \(0.610 \mathrm{~m} / \mathrm{s} .\) If a person stubs his toe, find (a) the time for each type of impulse to reach his brain, and (b) the time delay between the pain and touch impulses. Assume that his brain is \(1.85 \mathrm{~m}\) from his toe and that the impulses travel directly from toe to brain.

Galileo used marbles rolling down inclined planes to deduce some basic properties of constant accelerated motion. In particular, he measured the distance a marble rolled during specific time periods. For example, suppose a marble starts from rest and begins rolling down an inclined plane with constant acceleration \(a\). After 1 s, you find that it moved a distance \(x\). (a) In terms of \(x,\) how far does it move in the next 1 s time period-that is, in the time between \(1 \mathrm{~s}\) and \(2 \mathrm{~s} ?\) (b) How far does it move in the next second of the motion? (c) How far does it move in the \(n\) th second of the motion?

The Beretta Model \(92 \mathrm{~S}\) (the standard-issue U.S. army pistol) has a barrel \(127 \mathrm{~mm}\) long. The bullets leave this barrel with a muzzle velocity of \(335 \mathrm{~m} / \mathrm{s}\). (a) What is the acceleration of the bullet while it is in the barrel, assuming it to be constant? Express your answer in \(\mathrm{m} / \mathrm{s}^{2}\) and in \(g^{\prime}\) s. (b) For how long is the bullet in the barrel?

Cheetahs, the fastest of the great cats, can reach \(45 \mathrm{mi} / \mathrm{h}\) in \(2.0 \mathrm{~s}\) starting from rest. Assuming that they have constant acceleration throughout that time, find (a) their acceleration (in \(\mathrm{ft} / \mathrm{s}^{2}\) and \(\left.\mathrm{m} / \mathrm{s}^{2}\right)\) and \((\mathrm{b})\) the distance (in \(\mathrm{m}\) and \(\mathrm{ft}\) ) they travel during that time.

A jetliner has a cruising air speed of \(600 \mathrm{mi} / \mathrm{h}\) relative to the air. How long does it take this plane to fly round trip from San Francisco to Chicago, an east-west flight of \(2000 \mathrm{mi}\) each way, (a) if there is no wind blowing and (b) if the wind is blowing at \(150 \mathrm{mi} / \mathrm{h}\) from the west to the east?
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