# 0.3 Gravity and mechanical energy  (Page 9/9)

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## Potential energy

1. A tennis ball, of mass $120\phantom{\rule{2pt}{0ex}}g$ , is dropped from a height of $5\phantom{\rule{2pt}{0ex}}m$ . Ignore air friction.
1. What is the potential energy of the ball when it has fallen $3\phantom{\rule{2pt}{0ex}}m$ ?
2. What is the velocity of the ball when it hits the ground?
2. A bullet, mass $50\phantom{\rule{2pt}{0ex}}g$ , is shot vertically up in the air with a muzzle velocity of $200\phantom{\rule{2pt}{0ex}}m·s{}^{-1}$ . Use the Principle of Conservation of Mechanical Energy to determine the height that the bullet will reach. Ignore air friction.
3. A skier, mass $50\phantom{\rule{2pt}{0ex}}\mathrm{kg}$ , is at the top of a $6,4\phantom{\rule{2pt}{0ex}}m$ ski slope.
1. Determine the maximum velocity that she can reach when she skies to the bottom of the slope.
2. Do you think that she will reach this velocity? Why/Why not?
4. A pendulum bob of mass $1,5\phantom{\rule{2pt}{0ex}}\mathrm{kg}$ , swings from a height A to the bottom of its arc at B. The velocity of the bob at B is $4\phantom{\rule{2pt}{0ex}}m·s{}^{-1}$ . Calculate the height A from which the bob was released. Ignore the effects of air friction.
5. Prove that the velocity of an object, in free fall, in a closed system, is independent of its mass.

## Energy graphs

Let us consider our example of the suitcase on the cupboard, once more.

Let's look at each of these quantities and draw a graph for each. We will look at how each quantity changes as the suitcase falls from the top to the bottom of the cupboard.

• Potential energy : The potential energy starts off at a maximum and decreases until it reaches zero at the bottom of the cupboard. It had fallen a distance of 2 metres.
• Kinetic energy : The kinetic energy is zero at the start of the fall. When the suitcase reaches the ground, the kinetic energy is a maximum. We also use distance on the $x$ -axis.
• Mechanical energy : The mechanical energy is constant throughout the motion and is always a maximum. At any point in time, when we add the potential energy and the kinetic energy, we will get the same number.

The following presentation provides a summary of some of the concepts covered in this chapter.

## Summary

• Mass is the amount of matter an object is made up of.
• Weight is the force with which the Earth attracts a body towards its centre.
• Newtons Law of Gravitation.
• A body is in free fall if it is moving in the Earth's gravitational field and no other forces act on it.
• The equations of motion can be used for free fall problems. The acceleration (a) is equal to the acceleration due to gravity (g).
• The potential energy of an object is the energy the object has due to his position above a reference point.
• The kinetic energy of an object is the energy the object has due to its motion.
• Mechanical energy of an object is the sum of the potential energy and kinetic energy of the object.
• The unit for energy is the joule (J).
• The Law of Conservation of Energy states that energy cannot be created or destroyed, but can only be changed from one form into another.
• The Law of Conservation of Mechanical Energy states that the total mechanical energy of an isolated system remains constant.
• The table below summarises the most important equations:
 Weight ${F}_{g}=m·g$ Equation of motion ${v}_{f}={v}_{i}+gt$ Equation of motion $\Delta x=\frac{\left({v}_{i}+{v}_{f}\right)}{2}t$ Equation of motion $\Delta x={v}_{i}t+\frac{1}{2}g{t}^{2}$ Equation of motion ${v}_{f}^{2}={v}_{i}^{2}+2g\Delta x$ Potential Energy $PE=mgh$ Kinetic Energy $KE=\frac{1}{2}m{v}^{2}$ Mechanical Energy $U=KE+PE$

## End of chapter exercises: gravity and mechanical energy

1. Give one word/term for the following descriptions.
1. The force with which the Earth attracts a body.
2. The unit for energy.
3. The movement of a body in the Earth's gravitational field when no other forces act on it.
4. The sum of the potential and kinetic energy of a body.
5. The amount of matter an object is made up of.
2. Consider the situation where an apple falls from a tree. Indicate whether the following statements regarding this situation are TRUE or FALSE. Write only 'true' or 'false'. If the statement is false, write down the correct statement.
1. The potential energy of the apple is a maximum when the apple lands on the ground.
2. The kinetic energy remains constant throughout the motion.
3. To calculate the potential energy of the apple we need the mass of the apple and the height of the tree.
4. The mechanical energy is a maximum only at the beginning of the motion.
5. The apple falls at an acceleration of $9,8\phantom{\rule{2pt}{0ex}}m·s{}^{-2}$ .
3. [IEB 2005/11 HG] Consider a ball dropped from a height of $1\phantom{\rule{2pt}{0ex}}m$ on Earth and an identical ball dropped from $1\phantom{\rule{2pt}{0ex}}m$ on the Moon. Assume both balls fall freely. The acceleration due to gravity on the Moon is one sixth that on Earth. In what way do the following compare when the ball is dropped on Earth and on the Moon.
 Mass Weight Increase in kinetic energy (a) the same the same the same (b) the same greater on Earth greater on Earth (c) the same greater on Earth the same (d) greater on Earth greater on Earth greater on Earth
4. A man fires a rock out of a slingshot directly upward. The rock has an initial velocity of $15\phantom{\rule{2pt}{0ex}}m·s{}^{-1}$ .
1. How long will it take for the rock to reach its highest point?
2. What is the maximum height that the rock will reach?
3. Draw graphs to show how the potential energy, kinetic energy and mechanical energy of the rock changes as it moves to its highest point.
5. A metal ball of mass $200\phantom{\rule{2pt}{0ex}}g$ is tied to a light string to make a pendulum. The ball is pulled to the side to a height (A), $10\phantom{\rule{2pt}{0ex}}\mathrm{cm}$ above the lowest point of the swing (B). Air friction and the mass of the string can be ignored. The ball is let go to swing freely.
1. Calculate the potential energy of the ball at point A.
2. Calculate the kinetic energy of the ball at point B.
3. What is the maximum velocity that the ball will reach during its motion?
6. A truck of mass $1,2\phantom{\rule{2pt}{0ex}}\mathrm{tons}$ is parked at the top of a hill, $150\phantom{\rule{2pt}{0ex}}m$ high. The truck driver lets the truck run freely down the hill to the bottom.
1. What is the maximum velocity that the truck can achieve at the bottom of the hill?
2. Will the truck achieve this velocity? Why/why not?
7. A stone is dropped from a window, $3\phantom{\rule{2pt}{0ex}}m$ above the ground. The mass of the stone is $25\phantom{\rule{2pt}{0ex}}g$ .
1. Use the Equations of Motion to calculate the velocity of the stone as it reaches the ground.
2. Use the Principle of Conservation of Energy to prove that your answer in (a) is correct.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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