If your solution to stated problem does not match the given solution, you should self-critique per instructions at

 

   http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

.

Your solution, attempt at solution.  If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it.  This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

 

017.  `query 17

 

ANSWERS/COMMENTARY FOR QUERY 17

 

Question:  `qprin phy and gen phy 6.33:  jane at 5.3 m/s; how high can she swing up that vine?

 

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence rating:

Given Solution: 

Outline of solution:

Jane has KE.  She goes higher by increasing her gravitational PE.

Her KE is 1/2 m v_0^2, where m is her mass and v0 is her velocity (in this case, 6.3 m/s^2).  If she can manage to convert all her KE to gravitational PE, her KE will decrease to 0 (a decrease of 1/2 m v0^2) and her gravitational PE will therefore increase by amount 1/2 m v_0^2.

The increase in her gravitational PE  is m g `dy, where m is again her mass and `dy is the increase in her altitude.

Thus we have

PE increase = KE loss

In symbols this is written

 m g `dy = 1/2 m v0^2.

The symbol m stands for Jane's mass, and we can also divide both sides by m to get

g `dy = 1/2 v0^2.

Since we know g = 9.8 m/s^2 and v0 = 6.3 m/s, we can easily find `dy.

`dy = v0^2 / (2 g)

which is easily evaluated to obtain `dy = 1.43 m.

MORE DETAILED SOLUTION:

Jane is going to convert her KE to gravitational PE.  We assume that nonconservative forces are negligible, so that `dKE + `dPE = 0 and `dPE = -`dKE. 

Jane's KE is .5 M v^2, where M is her mass.  Assuming she swings on the vine until she comes to rest at her maximum height, the change in her KE is therefore  

Her change in gravitational PE is M g `dy, where `dy is the change in her vertical position.  So we have 

`dKE = - `dPE, or

- 1/2 M v0^2 = - ( M g `dy), which we solve for `dy (multiply both sides by -1, divide both sides by M g) to obtain

`dy = v0^2 / (2 g) = (5.3 m/s)^2 / (2 * 9.8 m/s^2) = 1.43 m.

 STUDENT QUESTION:

 I’m confused as to where the 2 g came from

INSTRUCTOR RESPONSE:

You are referring to the 2 g in the last line.

We have in the second-to-last line

- 1/2 M v0^2 = - ( M g `dy).  Dividing both sides by - M g, and reversing the right- and left-hand sides, we obtain

`dy = - 1/2 M v0^2 / (M g) = 1/2 v0^2 / g = v0^2 / (2 g).

STUDENT QUESTION

do we get dy'=v0^2/2g will this always be the case?

INSTRUCTOR RESPONSE

Most basic idea:

On the simplest level, this is a conversion of PE to KE. This is the first thing you should understand. 

The initial KE will change to PE, so the change in PE is equal to the initial KE. 

In this case the change in PE is m g `dy.  For other situations and other conservative forces the expression for `dPE will be very different.

The simplest equation for this problem is therefore

init KE = increase in PE so that

1/2 m v0^2 = m g `dy

More general way of thinking about this problem:

More generally we want to think in terms of KE change and PE change.  We avoid confusion by not worrying about whether each change is a loss or a gain.

Whenever conservative forces are absent, or being regarded as negligible, we can set the expression for KE change, plus the expression for PE change, equal to zero. 

-1/2 m v0^2 + m g `d y = 0. 

The very last step in setting up the problem should be to write out the expressions for KE and PE changes.

In this particular case the equation will read

If we let vf = 0, the previous equations follow.

 STUDENT QUESTION

Ok I understand gravitiaional PE=mgdy and I understand KE=.5mv^2
But Idont really understand how to combine the two equations. I can see that doing so allows us to eliminate m. since we don’t have that value.
And I don’t see how KE and PE can cancel out to allow for the equations to combine.



INSTRUCTOR RESPONSE

In the absence of dissipative forces (e.g., friction) it is possible to convert KE to PE, or vice versa.

Jane is moving at 5.3 m/s so she has KE.

She wants to go higher so she converts her KE to PE by grabbing a vine. As she follows the resulting upward ard her PE increases at the expense of here KE. She continues rising until she has used up all her KE.

At that point her KE will have been converted to PE.

If her altitude increases by `dy, her PE will have increased by m g `dy.

Her original KE was 1/2 m v0^2, where v0 is her original 5.3 m/s velocity.

So we set m g `dy equal to 1/2 m v0^2, and solve for `dy.

RESPONSE IN TERMS OF WORK-ENERGY THEOREM

In this case, Jane's kinetic energy is converted to gravitational potential energy.

Her kinetic energy is 1/2 m v^2 and her gravitational potential energy is m g `dy. In the simplest terms, the two will therefore be equal, with 1/2 m v^2 = m g `dy.

In terms of the work-energy theorem, we assume no loss to friction or other non-conservative forces so `dW_NC = 0. Since

`dW_NC = `dKE + `dPE,

we conclude that

`dKE + `dPE = 0

so that

`dPE = - `dKE.

Her kinetic energy will decrease from 1/2 m v^2 to 0, a change of - 1/2 m v^2. So

`dPE = - (-1/2 m v^2) = 1/2 m v^2

Since `dPE = m g `dy, we have

m g `dy = 1/2 m v^2.
 

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Question:  `qprin phy and gen phy 6.39:  950 N/m spring  compressed .150 m, released with .30 kg ball.  Upward speed, max altitude of ball

 

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence rating::

Given Solution: 

`a

We being with a few preliminary observations:

The PE stored in the spring will be .5 k x^2 = .5 ( 950 N/m ) ( .150 m)^2 = 10.7 J. 

Since the ball is moving in the vertical direction, between the release of the spring and the return of the spring to its equilibrium position, the ball has a change in gravitational PE as well as elastic PE. 

Summarizing what we know so far: 

 

During this interval, the KE change of the ball must therefore be `dKE = - `dPE = - (-10.3 J) = +10.3 J. 

Intuitively, the ball gains in the form of KE the 10.3 J of PE lost by the system.

 

The initial KE of the ball is 0, so its final KE during its interval of contact with the spring is 10.3 J.  We therefore have

To find the max altitude to which the ball rises, we consider the interval between release of the spring and maximum height. 

`dy = `dPE / (m g)

= 10.7 J / ( .30 kg * 9.8 m/s^2)

= 10.7 J / (2.9 N) = 10.7 N * m / (2.9 N) = 3.7 meters.

 SOLUTION BY DETAILED USE OF WORK-KINETIC ENERGY THEOREM

If there are no nonconservative forces, then `dW_NC = 0 so that the work-energy theorem

`dW_NC_ON = `dKE + `dPE

gives us

`dKE + `dPE = 0.

This will be the can whenever nonconservative forces are absent (of negligible).

In the preceding problem with Jane and the grapevine, the PE change was gravitational, as her swing on the vine elevated her higher than her initial position.

In this problem the spring releases its PE, which gives the ball enough KE to rise to some height.

It would be somewhat complicated to analyze the details of this motion, but all we're asked is how high the ball rises, so we can bypass all those details by realizing that the ball has the same initial and final velocity, which is zero. So its KE at release and at maximum height is zero. There is therefore no net change in KE.

As a result, we have `dKE = 0.

Since `dKE + `dPE = 0 and `dKE = 0, we conclude that `dPE = 0.

 

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Question (optional Openstax for 'prin and 'gen): 

(a) How high a hill can a car coast up (engine disengaged) if work done by friction is negligible and its initial speed is 110 km/h?

(b) If, in actuality, a 750-kg car with an initial speed of 110 km/h is observed to coast up a hill to a height 22.0 m above its starting point, how much thermal energy was generated by friction?

(c) What is the average force of friction if the hill has a slope 2.5 degrees above the horizontal?

[The normal force on the car when on a 2.5 degree incline is 99.5% of its weight, which you should verify if at this point of the course you know how.

The 'rise' of a 2.5 degree incline is about 4% of the distance traveled along the incline.  This is easily verified using trigonometry, but if you haven't yet learned the trigonometry you can use this approximation.  If you know the trigonometry, you should use it in your solution.]

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence rating::

Given Solution: 

Overall idea:

The car will dissipate its kinetic energy as it runs up the hill, thereby increasing its gravitational potential energy as friction does negative work on it.  It thus 'spends' its initial KE on the PE increase and the work it does against friction.

Key formulation:

`dW_noncons_ON = `dKE + `dPE,

the work done ON the car by nonconservative forces is equal the net change in its mechanical energy (mechanical energy being the sum of its kinetic and potential energies).

Important special case:

If `dW_noncons_ON = 0 then `dKE + `dPE = 0, meaning that any change in kinetic energy is associated with an equal and opposite change in potential energy.

Solution:

(a) The car will coast up the hill until its kinetic energy has been partially dissipated against friction and partially converted to gravitational PE.  If friction is negligible, then the KE will all be converted to gravitational PE.

No mass is given for the car, so we will let m_car stand for its mass.

The initial KE of the car is

KE_0 = 1/2 m_car v_0^2

= 1/2 m_car (110 km / h)^2

= 1/2 m_car ( 110 (1000 m) / (3600 s) ) ^2

= 1/2 m_car * (30 m/s)^2

= m_car * 450 m^2 / s^2,

where the 30 m/s is a rough approximation of 110 (1000 m) / (3600 s).

We understand that gravitational PE increases with altitude.  To take this back to the definition: 

The change in gravitational PE is equal and opposite to the work done by the gravitational force.  Gravity exerts a downward force of magnitude m_car g, so if the car's vertical displacement is upward, gravity will do negative work and the PE change is therefore positive.  If the upward direction is regarded as positive, then the force exerted by gravity (i.e., the weight of the car) is in the negative direction so will be represented by -m_car g. 

if the change in vertical position is `ds_y, then the work done by gravity is -(m_car g) * `ds_y and the change in gravitational PE is + m_car g `ds_y.

The KE loss is therefore 1/2 m_car v_0^2 = m_car * 450 m^2 / s^2, which is equal to the increase in gravitational PE.  So we can write

m_car g `ds_y = m_car * 450 m^2 / s^2.

Substituting 9.8 m/s^2 for g, we easily solve this equation for `ds_y and obtain `ds_y = 46 meters.

It is preferable to also write and solve the same equation symbolically:

m_car g `ds_y = 1/2 m_car v_0^2.

Solving for v we first divide both sides by m_car to get

g `ds_y = 1/2 v_0^2  so that

`ds_y = 1/2 v_0^2 / g = 1/2 * (30 m/s)^2 / (9.8 m/s^2) = 46 (m^2 / s^2) / (m/s^2) = 46 (m^2 / s^2) ( s^2 / m) = 46 m.

(b) If the car coasts to a point just 22 m higher than the original, then the change in its gravitational PE is

`dPE = m_car * g * `ds_y = 750 kg * 9.8 m/s^2 * 22 m = 160 000 kg m^2 / s^2 = 160 000 Joules.

Its initial KE was

KE_0 = 1/2 m v_0^2 = 1/2 * 750 kg * (30 m/s)^2 = 340 000 Joules, approx.

Its KE therefore changed by -340 000 Joules while its PE increased by 160 000 Joules.  Intuitively, were no friction present the 340 000 Joules loss of KE would be accompanied by a 340 000 Joule increase in PE, but the PE change comes up 180 000 Joules short.  This 180 000 Joules of energy was therefore dissipated by friction.

In terms of the work-energy theorem:

`dW_noncons_ON = `dKE + `dPE = -340 000 Joules + 160 000 Joules = -180 000 Joules.

The nonconservative force at work here is friction ON the car, so `dW_noncons_ON = -180 000 Joules.  That is, friction does -180 000 Joules of work on the car.

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Question:  `qgen phy  problem A high jumper needs to be moving fast enough at the jump to lift her center of mass 2.1 m and cross the bar at a speed of .7 m/s.  What minimum velocity does she require?

 

Given Solution: 

`aFORMAL SOLUTION:

 

Formally we have `dPE + `dKE = 0.

 

`dPE = M * g * `dy = M * 20.6 m^2 / sec^2, where M is the mass of the jumper and `dy is the 2.1 m change in altitude.

 

`dKE = .5 M vf^2 - .5 M v0^2, where vf is the .7 m/s final velocity and v0 is the unknown initial velocity.

 

So we have

 

M g `dy + .5 M vf^2 - .5 M v0^2 = 0.

 

Dividing through by M we have

 

g `dy + .5 vf^2 - .5 v0^2 = 0.

 

Solving for v0 we obtain

 

v0 = sqrt( 2 g `dy + vf^2) = sqrt( 2* 9.8 m/s^2 * 2.1 m + (.7 m/s)^2 ) = sqrt( 41.2 m^2/s^2 + .49 m^2 / s^2) = sqrt( 41.7 m^2 / s^2) = 6.5 m/s, approx..

 

LESS FORMAL, MORE INTUITIVE, EQUIVALENT SOLUTION:

 

The high jumper must have enough KE at the beginning to increase his PE through the 2.1 m height and to still have the KE of his .7 m/s speed.

 

The PE change is M * g * 2.1 m = M * 20.6 m^2 / sec^2, where M is the mass of the jumper

 

The KE at the top is .5 M v^2 = .5 M (.7 m/s)^2 = M * .245 m^2 / s^2, where M is the mass of the jumper.

 

Since the 20.6 M m^2 / s^2 increase in PE must come at the expense of the initial KE, and since after the PE increase there is still M * .245 m^2 / s^2 in KE, the initial KE must have been 20.6 M m^2 / s^2 + .245 M m^s / s^2 =20.8 M m^s / s^2, approx.

 

If initial KE is 20.8 M m^s / s^2, then .5 M v0^2 = 20.8 M m^s / s^2.

 

We divide both sices of this equation by the jumper's mass M to get

 

.5 v0^2 = 20.8 m^2 / s^2, so that

 

v0^2 = 41.6 m^2 / s^2 and

 

v0 = `sqrt(41.6 m^2 / s^2) = 6.5 m/s, appprox.

STUDENT QUESTION

I used the equation 'dy=v0^2 / (2g).  Isn't that easier?

INSTRUCTOR RESPONSE

Good, but that equation only applies under certain conditions. Your solution didn't account for the final KE, which doesn't make a lot of difference but does make enough to decide the winner of a competitive match.

In general you don't want to carry an equation like 'dy=v0^2/(2g) around with you. If you carry that one around, there are about a hundred others that apply to different situations, and you'll overload very quickly. Among other things, that equation doesn't account for both initial and final KE. It applies only when the PE change is gravitational, only near the surface of the Earth, and only when the final KE is zero. Way too many special conditions to keep in mind, way too much to remember.

You want to start your reasoning from `dKE + `dPE + `dW_noncons_ON = 0.

We assume that nonconservative forces are negligible, so that `dW_noncons_ON is itself zero, giving us

`dPE + `dKE = 0.

For this situation `dPE = m g `dy, `dKE = KE_f - KE_0 = 1/2 m vf^2 - 1/2 m v0^2, and the equation becomes

m g `dy + 1/2 m v0^2 - 1/2 m vf^2 = 0.

In a nutshell, there are only three things you need in order to analyze similar situations: 

STUDENT QUESTION

After looking throught the given solution I don’t see how any of the values were obtained.
The only ones I understand is the dy=2.1 m and vf=.7 m/s
Again I think a major problem for at this point is that I do not understand how to combine the equations. Ive read all of the notes and problem sets etc. However maybe I just didn’t take note of someof the things I should’ve at the time. If you tell me where I could look for that I think that would help me. Thanks.

Your text should provide a good reference, as will the Introductory Problem Sets and qa's..

This problem is similar to the earlier problem. You can figure out the PE change as a multiple of the jumper's mass, using `dy = 2.1 m and g.= 9.8 m/s^2.

You can figure out the jumper's KE as she goes over the bar at .7 m/s, again as a multiple of her mass.

To get to the height of the bar and the .7 m/s velocity, she has to have KE equal to the sum of the PE gain and the KE at .7 m/s.

So set 1/2 m v0^2 equal to the sum of the other two expressions and solve for v0.
 

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Question (optional Openstax `prin, `gen): 

Using energy considerations and assuming negligible air resistance, show that a rock thrown from a bridge 20.0 m above water with an initial speed of 15.0 m/s strikes the water with a speed of 24.8 m/s independent of the direction thrown.

 Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence rating::

Given Solution: 

Main idea:

The rock starts kinetic energy.  Gravity does positive work on it as it falls, decreasing its potential energy and therefore increasing its velocity.  Friction doesn't hinder the process.

Formulation:

`dW_noncons_ON = `dKE + `dPE.

In this case the only nonconservative force would be friction, which however is assumed to be zero so that `dW_noncons_ON = 0.  It follows that `dKE + `dPE = 0.  Any change in KE or PE is accompanied by an equal and opposite change in the other.

Solution:

Gravity does positive work on the rock so its change in gravitational PE, being equal and opposite to the work done by gravity, is negative. 

We get

`dPE = m_rock * g * `ds_y = m_rock * 9.8 m/s^2 * (-20 m) = -m_rock * 196 m^2 / s^2.

It follows that `dKE = +m_rock * 196 m^2 / s^2, so that

KE_f = KE_0 + `dKE =

1/2 m_rock * (24.8 m/s)^2 + m_rock * 196 m^2 / s^2 =

m_rock * 310 m^2 / s^2 + m_rock * 196 m^2 / s^2 =

m_rock *  500 m^2 / s^2, approximately.

KE_f = 1/2 m_rock vf^2, so

1/2 m_rock vf^2 = m_rock * 500 m^2 / s^2 and

vf = +- sqrt( 2 * 500 m^2 / s^2 ) = +-31 m/s, approx.

A purely symbolic solution is more elegant:

`dW_noncons_ON = 0 so, since `dW_noncons_ON = `dPE + `dKE we have

`dKE + `dPE = 0

For this situation

`dKE = 1/2 m_rock v_f^2 - 1/2 m_rock v0^2

`dPE = m_rock g `ds_y.

Substituting into `dKE + `dPE = 0 we have

1/2 m_rock v_f^2 - 1/2 m_rock v0^2 +  m_rock g `ds_y = 0

We divide both sides by m_rock to get

1/2 v_f^2 - 1/2 v0^2 +  g `ds_y = 0

and solve for v_f to get

vf = +- sqrt( 2 (1/2 v0^2 - g `ds_y)).

Substituting for v0, g and `ds_y we have

vf = +- sqrt( 2 * (1/2 * (24.8 m/s)^2 - 9.8 m/s^2 * (-20 m) )

= +- sqrt( 600 m^2 / s^2 + 400 m^2 / s^2)

= +- sqrt( 1000 m^2 / s^2) = +- 31 m/s, approximately.

We just want the speed of the rock, do we ignore the negative solution and conclude that the final speed is about

vf = 31 m/s.

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Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence rating:

Question:  `qquery Univ.  7.42 (7.38 in 10th edition).  2 kg block, 400 N/m spring, .220 m compression.  Along surface then up 37 deg incline all frictionless.  How fast on level, how far up incline?

 

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence rating::

Given Solution: 

`a** The spring exerts a force of 400 N / m * .220 m = 88 N at the .220 m compression.  The average force exerted by the spring between equilibrium and this point is therefore (0 N + 88 N) / 2 = 44 N, so the work done in the compression is

 

`dW = Fave * `ds = 44 N * .220 m = 9.7 Joules, approx. 

 

If all this energy is transferred to the block, starting from rest, the block's KE will therefore be 9.7 Joules.  Solving KE = .5 m v^2 for v we obtain v = sqrt(2 KE / m) = sqrt(2 * 9.7 Joules / (2 kg) ) = 3.2 m/s, approx..

 

No energy is lost to friction so the block will maintain this speed along the level surface.  As it begins to climb the incline it will gain gravitational PE at the expense of KE until the PE is 9.7 J and the KE is zero, at which point it will begin to slide back down the incline.

 

After traveling through displacement `ds along the incline the height of the mass will be `ds sin(37 deg) = .6 `ds, approx., and its gravitational PE will be PE = m g h = m g * .6 `ds = .6 m g `ds.

 

Setting this expression equal to KE we obtain the equation

 

.6 m g `ds = KE,

 

which we solve for `ds to obtain

 

`ds = KE / (.6 m g) = 9.7 Joules / (.6 * 2 kg * 9.8 m/s^2) = .82 meters, approx.  **

 

 

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Question:  `qquery univ phy  7.50 62 kg skier, from rest, 65 m high.  Frict does -10.5 kJ.

 

What is the skier's speed at the bottom of the slope?

 

After moving horizontally over 82 m patch, air res 160 N, coeff frict .2, how fast is she going?

 

Penetrating 2.5 m into the snowdrift, to a stop, what is the ave force exerted on her by the snowdrift?

 

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence rating::

Given Solution: 

`a** The gravitational PE of the skier decreases by 60 kg * 9.8 m/s^2 * 65 m = 38 kJ, approx. (this means 38 kiloJoules, or 38,000 Joules).  The PE loss partially dissipated against friction, with the rest converted to KE, resulting in KE = 38 kJ / 10.5 kJ = 27.5 kJ.

 

Formally we have

 

`dKE + `dPE + `dWnoncons = 0, where `dWnoncons is the work done by the skier against friction.  Since friction does -10.5 kJ of work on the skier, the skier does 10.5 kJ of work against friction and we have `dKE = -`dPE - `dWnoncons = - (-38 kJ) - 10.5 kJ = 27.5 kJ.

 

The speed of the skier at this point will be

 

v = sqrt( 2 KE / m) = sqrt( 2 * 27,500 J / (65 kg) ) = 30 m/s, approx.

 

Over the 82 m patch the force exerted against friction will be .2 * 60 kg * 9.8 m/s^2 = 118 N, approx., so the force exerted against nonconservative forces will be 118 N + 160 N = 280 N approx..  The work done will therefore be

 

`dWnoncons = 280 N * 82 m = 23 kJ, approx.,

 

and the skier's KE will be

 

KE = 27.5 kJ - 23 kJ = 4.5 kJ, approx.

 

This implies a speed of

 

v = sqrt( 2 KE / m) = 12 m/s, approx.

 

To stop from this speed in 2.5 m requires that the remaining 4.5 kJ of KE be dissipated in the 2.5 m distance.  Thus we have

 

`dW = Fave * `ds, so that

 

Fave = `dW / `ds = 4500 J / (2.5 m) = 1800 N.

 

This is a significant force, about 3 times the weight of the skier, but distributed over a large area of her body will cause a good jolt, but will not be likely to cause injury.**

STUDENT QUESTION

If the PE is = 20.6, then why is the initial KE= 20.6 so that we are adding the .245 to the initial KE of 20.6 to get
20.8, I thought that the KE was equal and opposite to the PE why would we not subtract here?

INSTRUCTOR RESPONSE

`dKE + `dPE = 0, provided there are no nonconservative forces acting on the system.

In such a case, PE goes up as the mass rises, so KE goes down. Another way of looking at it: All or part of the KE converts to PE.

The mass can only go as high as the initial KE permits. Once the initial KE is 'used up', no increase in PE is possible (recall the assumption that no nonconservative forces act during this phase of motion).

At maximum height the mass is still moving in the horizontal direction, so not all of the KE converts to PE.

In this case PE increases by 20.6 M m^2/s^2, .245 M m^2/s^2 of KE is still present at the highest point, so about 20.8 M m^2 / s^2 of KE must have been present initially.

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