Examkrackers Physics 326 Height of Two Balls Launched From Springs

[rowjimmy]

A 2 kg ball and an 8 kg ball are placed on separate springs, each with thesame spring constant. The springs are compressed by the same distance andreleased. Which of the following is true about the maximum heights reached bythe balls?

A. The 2 kg ball will go four times as high.

B. The 2 kg ball will go twice as high.

C. The balls will reach equal maximum heights.

D. The 8 kg ball will go four times as high.

Answer: The 2 kg ball will go twice as high.

My reasoning: the balls energy go from spring potential-->kinetic-->gravitationalpotential which is .5kx^2-->.5mv^2-->mgh. Because they both start offwith the same k and x they'll have the same potential energy from the spring.When they both reach the equilibrium point on the spring x = 0 and all theenergy is KE so .5kx^2 = .5mv^2 so the v = ((sqrt)kx^2/m) thus v isproportional to the square root of 1/m so as m quadruples v decreases by afactor of 2. When the balls are at their maximum height h = .5v^2/g so theheight is proportional to the velocity squared which is proportional to squareroot of 1/m so if m decreases by 1/4 v increases by 2 and the maximum heightincreases by a factor of four.

This is not what the answer is though . If thevelocity doubles when the mass is reduced by 1/4 and h is proportional to v^2then why wouldn't the height be increased by a factor of 4?

Exscamkrackers explanation: "All its kinetic energy changes topotential energy; potential energy is proportional to height, thus at halfwaythe energies will be equal. Find the maximum height from v=sqrt(2gh) and divideby 2."

What is "halfway" in this explanation? And how does dividing by 2somehow change the fact that v^2 is proportional to h? I thought linear functionalities(addition, subtraction, multiplication, and division) are ignored whendetermining proportionalities and the only consideration is non-linearfunctionalities (square roots, exponents, etc). For instance the height of aprojectile (not related to the above question) whose velocity is doubled wouldincrease by a factor of four because h=v^2/2g. Despite the fact that there is a2 in the denominator the proportionality does not consider it because it staysconstant.

Also a related subsequent question asked: if a 2kg and 8kg ball are placedon same spring at same time and compressed and released what will be theheights of the balls? Answer: The balls will reach equal maximum heights. Whydo the balls of different masses reach the same height in this situation butnot in the first? In both situations don't they have the same spring potentialenergy (PE= .5kx^2 (independent of mass)?

Thanks!!

---

Members don't see this ad.

 

[chiddler]

i'm looking at the book now; the solution is A not B.

1/2 kx^2 = mgh
h = 0.5kx^2/mg

h is in a simple inverse relationship with m so 4x mass = 4x less height.

 

[MedPR]

i'm looking at the book now; the solution is A not B.

1/2 kx^2 = mgh
h = 0.5kx^2/mg

h is in a simple inverse relationship with m so 4x mass = 4x less height.

+1. It helps when you read the correct answer correctly.

 

[rowjimmy]

Whoops!

But can you explain 328 (two balls launched from same spring)? How is it that they reach the same maximum heights? In both questions don't both balls have equivalent potential energies? If so, then how can they travel differing relative distances?

Thanks again!

 

[rowjimmy]

LOL! I know right....

 

[ljc]

Whoops!

But can you explain 328 (two balls launched from same spring)? How is it that they reach the same maximum heights? In both questions don't both balls have equivalent potential energies? If so, then how can they travel differing relative distances?

Thanks again!

Because the spring can't push two things at different speeds. They'll both leave with the same speed and thus go the same height. Think if you tried to throw a box straight up, with only a single piece of paper in it. Now try to throw that same box, but with a block of lead in it as well. The lead limits how fast you can accelerate the paper, no matter how light the paper is. They'll leave the box with the same speed, and thus reach the same height

 

[rowjimmy]

Because the spring can't push two things at different speeds. They'll both leave with the same speed and thus go the same height. Think if you tried to throw a box straight up, with only a single piece of paper in it. Now try to throw that same box, but with a block of lead in it as well. The lead limits how fast you can accelerate the paper, no matter how light the paper is. They'll leave the box with the same speed, and thus reach the same height

Thanks that was a pretty straightforward and intuitive explanation!