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Chapter 2: Problem 1

Another ball dropped from a tower A ball is again dropped from a tower of height \(h\) with initial velocity zero. Write a program that asks the user to enter the height in meters of the tower and then calculates and prints the time the ball takes until it hits the ground, ignoring air resistance. Use your program to calculate the time for a ball dropped from a \(100 \mathrm{~m}\) high tower.

Short Answer

Expert verified
The time taken for a ball to fall from 100 meters is approximately 4.52 seconds.

Step by step solution

01

Understand the formula

To find the time it takes for a ball to fall from a height, use the equation of motion for free fall: \[ t = \sqrt{\frac{2h}{g}} \] where: * \( t \) is the time in seconds * \( h \) is the height in meters * \( g \) is the acceleration due to gravity, which is approximately \( 9.8 \) m/s².
02

Write the Program Structure

Write a basic structure of the program that will ask the user to input the height and then calculate the time using the formula from Step 1.
03

Implement User Input

Implement code to take the height input from the user. Example in Python: ```pythonheight = float(input('Enter the height of the tower in meters: '))```
04

Calculate the Time

Calculate the time using the formula mentioned in Step 1. Make sure to import the math module for the square root function. Example in Python: ```pythonimport mathg = 9.8time = math.sqrt((2 * height) / g)```
05

Print the Result

Print the result to the user. Example in Python: ```pythonprint(f'The time taken for the ball to hit the ground is {time} seconds')```
06

Test the Program

Run the program and enter 100 meters as the height to test if the program correctly calculates the time.

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Key Concepts

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

headline of the respective core concept
When dealing with objects in free fall, the equation of motion is an essential tool. The general equation for free fall can be written as \[ t = \sqrt{\frac{2h}{g}} \]where:
  • \(t\) is the time in seconds
  • \(h\) is the height in meters
  • \(g\) is the acceleration due to gravity, approximately \(9.8\) m/s².
This equation can be derived from the basic kinematic equations for uniformly accelerated motion. For example, the equation \(s = ut + \frac{1}{2}at^2\) reduces to \(h = \frac{1}{2}gt^2\) when initial velocity \(u\) is zero and solving for \(t\) yields \(t = \sqrt{\frac{2h}{g}}\). This is fundamental for determining how long it takes for an object to fall from a certain height without considering air resistance.
headline of the respective core concept
Gravity is the force that pulls objects towards Earth. When an object is in free fall, gravity is the only force acting on it. This causes the object to accelerate at a constant rate of \(9.8\) m/s².
  • Gravity is one of the fundamental forces of nature.
  • This constant acceleration is denoted by the symbol \(g\).
  • On Earth, \(g\) is roughly \(9.8\) m/s².
Understanding gravity is crucial for free fall calculations as it is the driving force behind the equations of motion. In practical applications, if air resistance is considered, the calculated time will be longer, but in the scope of basic physics problems, such as our exercise, air resistance is often ignored for simplicity.
headline of the respective core concept
Python programming can be very useful for solving physics problems like free fall calculations. Here's how you can write a simple program in Python to compute the time taken for a ball to fall from a given height:
1. Ask the user to input the height of the tower. 2. Use the equation of motion to compute the time. 3. Print the computed time for the user.
Here's a step-by-step implementation:
    1. Import the Python math module as it contains mathematical functions we need. 2. Get user input for the height and convert it to a floating-point number. 3. Compute the time using the formula \(t = \sqrt{\frac{2h}{g}}\). 4. Print the result.
Example code: ```pythonimport mathheight = float(input('Enter the height of the tower in meters: '))g = 9.8time = math.sqrt((2 * height) / g)print(f'The time taken for the ball to hit the ground is {time} seconds')``` By practicing with such simple Python programs, students can better understand how computational tools can help solve physics problems efficiently.

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Most popular questions from this chapter

Prime numbers The program in Example \(2.8\) is not a very efficient way of calculating prime numbers: it checks each number to see if it is divisible by any number less than it. We can develop a much faster program for prime numbers by making use of the following observations: a) A number \(n\) is prime if it has no prime factors less than \(n\). Hence we only need to check if it is divisible by other primes. b) If a number \(n\) is non-prime, having a factor \(r\), then \(n=r s\), where \(s\) is also a factor. If \(r \geq \sqrt{n}\) then \(n=r s \geq \sqrt{n} s\), which implies that \(s \leq \sqrt{n}\). In other words, any non-prime must have factors, and hence also prime factors, less than or equal to \(\sqrt{n}\). Thus to determine if a number is prime we have to check its prime factors only up to and including \(\sqrt{n}\)-if there are none then the number is prime. c) If we find even a single prime factor less than \(\sqrt{n}\) then we know that the number is non-prime, and hence there is no need to check any further- we can abandon this number and move on to something else. Write a Python program that finds all the primes up to ten thousand. Create a list to store the primes, which starts out with just the one prime number 2 in it. Then for each number \(n\) from 3 to 10000 check whether the number is divisible by any of the primes in the list up to and including \(\sqrt{n}\). As soon as you find a single prime factor you can stop checking the rest of them- you know \(n\) is not a prime. If you find no prime factors \(\sqrt{n}\) or less then \(n\) is prime and you should add it to the list. You can print out the list all in one go at the end of the program, or you can print out the individual numbers as you find them.

Suppose arrays a and b are defined as follows: $$ \begin{aligned} &\text { from numpy import array } \\ &a=\operatorname{array}([1,2,3,4], \text { int }) \\ &b=\operatorname{array}([2,4,6,8], \text { int }) \end{aligned} $$ What will the computer print upon executing the following lines? (Try to work out the answer before trying it on the computer.) a) print \((b / a+1)\) b) print \((b /(a+1))\) c) print(1/a)

Planetary orbits The orbit in space of one body around another, such as a planet around the Sun, need not be circular. In general it takes the form of an ellipse, with the body sometimes closer in and sometimes further out. If you are given the distance \(\ell_{1}\) of closest approach that a planet makes to the Sun, also called its perihelion, and its linear velocity \(v_{1}\) at perihelion, then any other property of the orbit can be calculated from these two as follows. a) Kepler's second law tells us that the distance \(\ell_{2}\) and velocity \(v_{2}\) of the planet at its most distant point, or aphelion, satisfy \(\ell_{2} v_{2}=\ell_{1} v_{1}\). At the same time the total energy, kinetic plus gravitational, of a planet with velocity \(v\) and distance \(r\) from the Sun is given by $$ E=\frac{1}{2} m v^{2}-G \frac{m M}{r}, $$ where \(m\) is the planet's mass, \(M=1.9891 \times 10^{30} \mathrm{~kg}\) is the mass of the Sun, and \(G=6.6738 \times 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}\) is Newton's gravitational constant. Given that energy must be conserved, show that \(v_{2}\) is the smaller root of the quadratic equation $$ v_{2}^{2}-\frac{2 G M}{v_{1} \ell_{1}} v_{2}-\left[v_{1}^{2}-\frac{2 G M}{\ell_{1}}\right]=0 . $$ Once we have \(v_{2}\) we can calculate \(\ell_{2}\) using the relation \(\ell_{2}=\ell_{1} v_{1} / v_{2}\). b) Given the values of \(v_{1}, \ell_{1}\), and \(\ell_{2}\), other parameters of the orbit are given by simple formulas can that be derived from Kepler's laws and the fact that the orbit is an ellipse: $$ \begin{aligned} \text { Semi-major axis: } & a=\frac{1}{2}\left(\ell_{1}+\ell_{2}\right), \\ \text { Semi-minor axis: } & b=\sqrt{\ell_{1} \ell_{2}}, \\ \text { Orbital period: } & T=\frac{2 \pi a b}{\ell_{1} v_{1}}, \\ \text { Orbital eccentricity: } & e=\frac{\ell_{2}-\ell_{1}}{\ell_{2}+\ell_{1}} . \end{aligned} $$ Write a program that asks the user to enter the distance to the Sun and velocity at perihelion, then calculates and prints the quantities \(\ell_{2}, v_{2}, T\), and \(e\). c) Test your program by having it calculate the properties of the orbits of the Earth (for which \(\ell_{1}=1.4710 \times 10^{11} \mathrm{~m}\) and \(v_{1}=3.0287 \times 10^{4} \mathrm{~ms}^{-1}\) ) and Halley's comet \(\left(\ell_{1}=8.7830 \times 10^{10} \mathrm{~m}\right.\) and \(\left.v_{1}=5.4529 \times 10^{4} \mathrm{~m} \mathrm{~s}^{-1}\right)\). Among other things, you should find that the orbital period of the Earth is one year and that of Halley's comet is about 76 years.

Quantum potential step A well-known quantum mechanics problem involves a particle of mass \(m\) that encounters a one-dimensional potential step, like this: The particle with initial kinetic energy \(E\) and wavevector \(k_{1}=\sqrt{2 m E} / \hbar\) enters from the left and encounters a sudden jump in potential energy of height \(V\) at position \(x=0\). By solving the Schrödinger equation, one can show that when \(E>V\) the particle may either (a) pass the step, in which case it has a lower kinetic energy of \(E-V\) on the other side and a correspondingly smaller wavevector of \(k_{2}=\sqrt{2 m(E-V)} / \hbar\), or \((b)\) it may be reflected, keeping all of its kinetic energy and an unchanged wavevector but moving in the opposite direction. The probabilities \(T\) and \(R\) for transmission and reflection are given by $$ T=\frac{4 k_{1} k_{2}}{\left(k_{1}+k_{2}\right)^{2}}, \quad R=\left(\frac{k_{1}-k_{2}}{k_{1}+k_{2}}\right)^{2} . $$ Suppose we have a particle with mass equal to the electron mass \(m=9.11 \times\) \(10^{-31} \mathrm{~kg}\) and energy \(10 \mathrm{eV}\) encountering a potential step of height \(9 \mathrm{eV}\). Write a Python program to compute and print out the transmission and reflection probabilities using the formulas above.

A spaceship travels from Earth in a straight line at relativistic speed \(v\) to another planet \(x\) light years away. Write a program to ask the user for the value of \(x\) and the speed \(v\) as a fraction of the speed of light \(c\), then print out the time in years that the spaceship takes to reach its destination (a) in the rest frame of an observer on Earth and (b) as perceived by a passenger on board the ship. Use your program to calculate the answers for a planet 10 light years away with \(v=0.99 c\).
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