Conservation of Energy: Calculating Speeds and Heights in Physics Problems

let's work on this problem a block slides down a 150 meter inclined plane as shown in the picture below starting

from rest what is the speed of the block when it reaches the bottom of the

incline so we're going to use conservation of energy to solve this problem

so the initial mechanical energy has to equal the final mechanical energy the only force is acting on a block

are conservative forces like gravity so mechanical energies can serve

at point a the only form of energy that we have is potential energy and at point b

choosing this as the ground level there's no potential energy at point b however the block will have kinetic

energy at point b so we can set these two equal to each other

as the block slides down potential energies being converted to kinetic energy

the potential energy of the block is mgh and the kinetic energy of the block is one half mv squared

so we could cancel m therefore we don't need the mass of the block

now i'm going to multiply both sides by two so on the left i have two g h

is equal to two times a half these will cancel so that's one and that's going to equal v squared

taking the square root of both sides the final speed is going to be the square root of 2 g h

so let's go ahead and plug everything that we have so it's 2 times the gravitational

acceleration of 9.8 times the height of the block which is 150 meters high

so by the way this is the height so that's 150 meters so if you type this in your calculator

you should get 54.22 meters per second

so without friction that's how fast this block is going to be moving when it reaches position b

number two an 8 kilogram block compresses a horizontal spring

by 2.5 meters beyond its natural length as shown in a figure below what is the speed of the block as soon

as it's released from the spring so let's call this position a and at position b

when it's still on level ground we want to find the speed as soon as it's released

so at position a the block spring system has potential energy elastic potential energy energy

is stored in the spring and once it's released that energy is going to be converted to kinetic energy

now the height doesn't change so there's no gravitational potential energy so we can say

the elastic potential energy will be equal to the kinetic energy of the block energy is going to be transferred from

the spring to the block the elastic potential energy is one-half kx squared

the kinetic energy is one-half mv squared if we multiply both sides by 2 we can get rid of the fraction

so now our goal is to solve for the speed so we have k which is 300

and x is the amount the spring is compressed by which is 2.5 meters

the mass of the block is eight so let's calculate the final speed 300 times 2.5 squared

is 18.75 and if we divide that by 8 that's going to be 234.375 and that's equal to v squared

now let's take the square root of both sides so the speed is going to be 15.31

meters per second so this is the answer to part a that's going to be the speed of the

block as soon as it's released at position b now part b

how high up the hill will the block go so let's see if it gets up to that height

how can we calculate h now at position b the block has kinetic energy

at position c where it comes to rest is no longer going to have any kinetic energy

the only energy is going to have is gravitational potential energy so what we need to do

is set the kinetic energy equal to the gravitational potential energy

so this is going to be one half mv squared and that's going to equal mgh so once again we could cancel m

now let's go ahead and plug in everything that we have so it's one half times v squared which is

15.31 squared and that's equal to g times h so 15.31 squared

times 0.5 divided by 9.8 will give us a height of about

12 meters if you rounded the nearest whole number so that's how high

the block is going to go before it comes to rest number three

a 10 kilogram block slides down a hill that is 200 meters tall as shown in the figure below

with an initial speed of 12 meters per second what is the speed of the block when it

reaches the bottom of the hill at let's say position b assuming there's no friction

so at position a let's call that the original position the block has potential energy because

it's above position b it has the capability of falling

now because the block is in motion it has kinetic energy at position b it's only going to have

kinetic energy so at position a we have potential energy

and kinetic energy at position b we only have kinetic energy

so all of the potential energy at position a is going to go to the kinetic energy of

the object the object speed will increase beyond 12 meters per second so let's go ahead and

calculate that speed so this is going to be mgh plus one-half

mv initial squared so the initial speed is 12. we're looking for the final speed and that's

equal to the final kinetic energy of one-half v final square

so we could divide everything by m and multiply everything by two so it's going to be 2g h

plus v initial squared and that's equal to v final squared so basically that looks

like this equation v final squared is equal to v initial squared plus 2ad that's the formula that you've seen in

kinematics so let's calculate the final speed so v final squared is equal to the

initial squared which is 12 squared plus 2 times the gravitational acceleration

times the height of 200. so it's 12 squared plus 2 times 9.8 times 200 and that's

4064 and then take the square root of that number

and you should get a final speed of point 63.7 five meters per second so this is the speed of the object at

position b if there's no friction now let's move on to part b

what is the final speed of the block at position b after traveling a total distance of 500

meters given a coefficient of kinetic friction of 0.21 between a block and a surface

now i do want to modify this so let's say that the distance is only for

the horizontal portion of the incline so let's say that distance is 500 meters

so not the the total part so let's say this portion is frictionless

and the 500 meters is only for the horizontal portion of the surface knowing that how can we calculate the

final speed of the block a friction only acts on the bottom of the incline

so how can we modify this equation at position a we're still going to have potential and

kinetic energy now once you add friction to the mix the final speed will no longer be 63.75 it's

going to be less so some of the energy from the left side is not all going to go into kinetic

energy some of it is going to go into the work done by friction now if you put the work done by friction

on the left side it's going to be negative w because friction is going to decrease

the total energy of the object and if it's negative w on the left side then it's positive w on the right side

so this represents the work done by friction friction's job

is to take away the mechanical energy of the object and to convert it to thermal energy

so the energy loss from the object is going to be w and that amount of energy is going to be

converted to heat energy so let's go ahead and calculate w first so the work done by friction

is equal to the kinetic frictional force times the distance that the force acts over

distance and displacement are the same if the object doesn't change direction so this work is force times displacement

now the frictional force is equal to mu k times the normal force so we have the coefficient of kinetic

friction and the normal force on a horizontal surface

is mg now if this part had friction that would complicate the problem

because the normal force on let's say an incline is mg cosine theta and this is not a

straight incline so the angle changes so that's just going to make it more complicated

so let's keep the problem simple so now let's go ahead and calculate the work

done by friction so it's mu k which is 0.21 multiplied by the mass of the object

which is 10 times g times the distance that the object acts

over which is 500 meters so let's multiply 0.21 by 10

by 9.8 and by 500 so you should get 10

290 joules so that's the work done by friction now let's use that to calculate the

final kinetic energy and also the final speed so the potential energy is mgh

and the kinetic energy at position a is one half mv initial squared

and the final kinetic energy is one half and v final squared and then plus w

so let's plug in the values that we have so the potential energy is going to be 10

times 9.8 times a height of 200 and the initial kinetic energy is going

to be one half times 10 times the initial speed which is

12 meters per second and then that's going to be the final kinetic energy

plus the work done by friction which we could replace that with

10 290 joules so 10 times 9.8 times 200 that's 19 600 joules

so there's enough potential energy to satisfy the work done by friction

which tells us that the final speed is going to be greater than 12.

if this was less than friction the final speed will be less than 12. now let's multiply 0.5 by 10 by 12

squared so the kinetic energy of the object is very low

it's 720. let me make sure that's right and yeah that's it so now let's add 19 600

plus 720 and let's subtract it by 10290 so the final kinetic energy is ten

thousand and thirty joules so now let's set that equal to one half mv final squared

so we have one half times a mass of 10 times v final squared so half of 10 is 5 and 1030 divided by 5

is 2006 so the square of the final speed is 2006 now let's take the square root of that

number so the final speed is going to be 44.8 meters per

second which was less than the final speed in part a a 12 kilogram block moving at a speed of

15 meters per second crashes into a wall and comes to a complete stop

how much thermal energy was produced during the collision so let's draw a picture

so imagine you have a block that's sliding it's moving at a speed of 15 meters per second

and it's a 12 kilogram mass and then it collides with a wall

so once it hits the wall it comes to a stop so it's no longer moving where did all of the kinetic energy go

during collisions whenever you have an object in motion and then it's no longer in motion you

need to realize that the kinetic energy of that object was transformed into thermal energy it's

lost due to heat and that heat just radiates outward into the surroundings

think of let's say if you rub your hands quickly you're going to feel that your hands

start to get hot so all of that kinetic energy that your that your hands had as it was moving

back and forth a lot of it was converted to thermal energy and so heat was generated

and that's what's going to happen here when that block collides with the wall and both objects well the wall is not

moving but the block comes to a stop all of the kinetic energy that the block had was transferred to thermal energy

so let's calculate the initial kinetic energy of the block so it's one half mv squared

so that's one half times a mass of 12 kilograms

times the speed of 15 meters per second so half of 12 is 6 and fifteen squared is two twenty five

six times two twenty five is thirteen fifty so thirteen fifty

joules of kinetic energy was transformed into thermal energy so that's the answer

now let's move on to this problem a 1500 kilogram car moving east at 35 meters per second

crashes head on into another 1800 kilogram car moving west

at 30 meters per second causing both cars to come to a complete stop so let's turn this into a picture

so this is the 1500 kilogram object it's a car but i'm just going to draw a box

and it's moving east at 35 meters per second and then we have another

object which is 1800 kilograms and this vehicle is moving west at 30 meters per second

so they collide and they come to a complete stop so all of the kinetic energies of these

two objects that they once had all of that was converted

to thermal energy so to calculate the total thermal energy produced we need to calculate the total kinetic

energy which is the kinetic energy of the first object i'm going to call it object a

plus the kinetic energy of the subject excuse me the second object object b so let's

call this a and b so this is going to be one half mv squared plus one half

mv squared so the mass of the first object is 1500 and it has a speed of 35 meters per

second the mass of the second object is 1800 and it has a speed of 30 meters per

second so 0.5 times 1500 times 35 squared

that's 918 750 joules and then 0.5 times 1800 times 30 squared

is 810 000 joules so if we add these two numbers this will give us 1 million

728 000. seven hundred fifty joules so that's how much thermal energy was produced

during this collision consider this problem so we have a roller coaster that's

released from rest at point a how fast is it moving at point b

so how can we find the speed at point b we need to use conservation of energy at point a the roller coaster has

potential energy at point b all of that potential energy is converted to kinetic

so let's set the potential energy at point a equal to the kinetic energy of the roller coaster at point b

so we have mgh will equal one-half mv squared

so we could divide both sides by m and then if we multiply both sides by two

we're gonna have two g h is equal to one half times two is one so it's just gonna be v squared

so the velocity is going to be the square root of 2 g h so it's going to be the square root of 2

times 9.8 multiplied by the height which is 50.

so let's go ahead and plug this in so you should get 31.3

meters per second at point b so that's how fast the roller coaster is moving at point b if there's no

friction in this problem now no friction was mentioned so we're assuming there's no friction

now it looks like i forgot to write part b so we're going to move on to part c

how high is point c relative to point b and i forgot to give you the speed at point c at point c

the speed is 20 meters per second so with that information

calculate the height of the roller coaster at point c

point b is ground level so how high is that point c feel free to pause the video

so i'm going to focus on point a as my initial point so at point a all we had to begin with

was potential energy at point c the object is above ground level

so it has potential energy now it's still moving at point c so it also has kinetic energy

so we need to start with this expression so we have mgh this is going to be h initial

is equal to mgh final plus one half

mv final squared now we could divide every term by m and so we have this expression

gh is equal to gh final plus one half

v final squared so now let's plug in the numbers it's going to be 9.8

times the initial height which is 50 and that's equal to 9.8 times the final height which is

what we're looking for and then plus one half v final square v final at

point c is 20. so 9.8 times 50 that's 490

and so that's equal to 9.8 times h and 20 squared is 400 half of that is 200

now 490 minus 200 is 290. so let's take 290 and let's divide it by 9.8

so 29.59 is equal to h so that's how high it is

at point c now let's move on to part d how fast is the roller coaster moving

at point d so we're still going to use part i mean

point a as our initial point so at point a we still have initial

potential energy at point d we still have potential energy because

it's above ground level it has the ability to fall and at point d it's still moving so it

has kinetic energy so we're going to have the same equation mgh initial

is equal to mgh final plus one half

mv final squared and just like before we could cancel m so g is 9.8

h initial is still 50. now h final is now 15. so the difference between

part d and part c is that in part d we're looking for the final speed now so 9.8 times 50

that's still 490 and that's equal to 9.8 times 15 which is

147 now let's take 490 and subtract it by 147

so that's going to be 343 is equal to one half v final square now multiply both sides by two

so 343 times two is six hundred and eighty six one half times two is one so this is

equal to the square of the speed and now let's take the square root of

both sides so the square root of 686 is 26.19

meters per second so that's how fast the roller coaster is moving

at point d at point c was moving at 20 and at point

b was like 30 something so that's how you could solve the rollercoaster problem

using conservation of energy you