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Chapter 2: Problem 74

You toss a book into your dorm room, just clearing a windowsill \(4.2 \mathrm{m}\) above the ground. (a) If the book leaves your hand \(1.5 \mathrm{m}\) above the ground, how fast must it be going to clear the sill? (b) How long after it leaves your hand will it hit the floor, \(0.87 \mathrm{m}\) below the windowsill?

Short Answer

Expert verified
The initial speed required to clear the windowsill is given by equation in Step 1 (a) from the solution steps. The time it takes for the book to hit the floor is given by equation in Step 1 (b) from the solution steps.

Step by step solution

01

Identify and list down knowns

For both parts, let the initial position be the point at which the book leaves your hand. Let upwards be positive. So, the known quantities are: \n1. initial height = 1.5m \n2. height of windowsill = 4.2m \n3. height of the floor from the windowsill = 0.87m.\n4. acceleration due to gravity (g) = 9.8 m/s^2, towards downwards, hence -g.
02

for (a): Calculate initial speed

The equation for kinematics to find initial speed is \[v_{i}=\sqrt{2 g y_{max}}\] where \(v_{i}\) is the initial speed, \(g\) is acceleration due to gravity and \(y_{max}\) is the maximum height.\nHere, the maximum height is the height of the windowsill and initial position, so add the initial height (1.5m) and the height of windowsill (4.2m) to get \(y_{max} = 4.2 + 1.5 = 5.7m\).\nSubstitute this value and known value of \(g\) into the equation to solve for initial speed.
03

for (b): Calculate total flight time

The total flight time can be calculated with the kinematic equation \[t=\sqrt{\frac{2 h}{g}}\] where \(t\) is time, \(h\) is height and \(g\) is acceleration due to gravity.\nThe total height \(h\) is the addition of the initial height (1.5m), the height of windowsill (4.2m) and the height of the floor from the windowsill (0.87m).\nSo \(h = 1.5 + 4.2 + 0.87 = 6.57m\).\nSubstitute this value and known value of \(g\) into the equation to solve for time.

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Key Concepts

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

Projectile motion
When discussing projectile motion, we're referring to any object that is thrown or projected into the air and is only influenced by gravitational forces. In simpler terms, think about when you throw a ball in the air; what you observe is projectile motion. The ball moves in a curved path due to the force of gravity acting upon it. It's important to break down this motion into two components: horizontal and vertical.
  • The horizontal motion occurs at a constant speed as there is no acceleration involved (ignoring air resistance).
  • The vertical motion is impacted by gravity, causing the object to accelerate downward.

Understanding these components allows us to predict the path an object will take, such as the flight path of a book being tossed across a room. Each movement aids in calculating critical aspects like how far and how high the object will travel.
Initial speed calculation
Calculating the initial speed is crucial when determining how fast an object needs to be thrown to reach a certain height or distance. In our exercise, we want to know how speedy the book must be to clear the windowsill. To find this, we use a kinematic equation:
\[ v_{i} = \sqrt{2 g y_{max}} \]
Where:
  • \(v_i\) is the initial velocity we're looking to find.
  • \(g\) is the acceleration due to gravity, which is approximately \(9.8\, \text{m/s}^2\) on Earth.
  • \(y_{max}\) is the maximum height the object needs to reach, adding the height from which the object was released.
In this problem, we determined that the windowsill is 5.7 meters from where the book starts, considering all heights involved. So, when we substitute this height along with gravity into the equation, it gives us the initial speed needed for the book to clear the windowsill.
Free fall motion
Free fall motion is purely due to the force of gravity acting on an object. No other forces, like air resistance, are considered in these calculations. In our context, after the book clears the windowsill, it enters a free fall phase before hitting the floor. During free fall, the only force acting upon it is gravity pulling it down. Therefore, the object accelerates downward at a rate of \(9.8\, \text{m/s}^2\).
  • Gravity pulls objects towards the Earth at a constant acceleration.
  • This free fall continues until the object encounters an obstacle or the ground.
Understanding this concept helps you visualize how the book behaves once it finishes the upward journey and starts descending, moving faster as time progresses.
Time of flight
The time of flight refers to how long an object remains in the air from the moment it is released until it hits the ground. For the book exercise, we are interested in finding out when it will land on the floor below the sill. Utilizing the kinematic equation:
\[ t = \sqrt{\frac{2 h}{g}} \]
Where:
  • \(t\) is the total time the book spends in the air.
  • \(h\) is the total distance the book travels vertically.
  • \(g\) remains Earth's gravitational pull at \(9.8\, \text{m/s}^2\).
In our problem, \(h\) is calculated by adding the initial height, the windowsill height, and the distance to the floor. With a sum of 6.57 meters, we can substitute this value, alongside gravity, into the equation to find the time of flight. This helps predict exactly when the book will land after being tossed.

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Most popular questions from this chapter

The standard 26 -mile, 385 -yard marathon dates to \(1908,\) when the Olympic marathon started at Windsor Castle and finished before the Royal Box at London's Olympic Stadium. Today's top marathoners achieve times around 2 hours, 4 minutes for the standard marathon. (a) What's the average speed of a marathon run in this time? (b) Marathons before 1908 were typically about 25 miles. How much longer does the race last today as a result of the extra mile and 385 yards, assuming it's run at the average speed?

You allow 40 min to drive 25 mi to the airport, but you're caught in heavy traffic and average only \(20 \mathrm{mi} / \mathrm{h}\) for the first 15 min. What must your average speed be on the rest of the trip if you're to make your flight?

A jetliner leaves San Francisco for New York, \(4600 \mathrm{km}\) away. With a strong tailwind, its speed is \(1100 \mathrm{km} / \mathrm{h}\). At the same time, a second jet leaves New York for San Francisco. Flying into the wind, it makes only \(700 \mathrm{km} / \mathrm{h}\). When and where do the two planes pass?

A hockey puck moving at \(32 \mathrm{m} / \mathrm{s}\) slams through a wall of snow \(35 \mathrm{cm}\) thick. It emerges moving at \(18 \mathrm{m} / \mathrm{s}\). Assuming constant acceleration, find (a) the time the puck spends in the snow and (b) the thickness of a snow wall that would stop the puck entirely.

You check your odometer at the beginning of a day's driving and again at the end. Under what conditions would the difference between the two readings represent your displacement?
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