Conservation of Mechanical Energy

Important points to consider:  In the absence of friction, no energy is lost. It is only converted from one form to another. For example, energy may be converted from potential to kinetic and vice versa.

Pay attention to units. In the S.I. System, distance is in meter, and mass is in kilogram, while energy is in Joule.

The Problem:

A 180-gram baseball is launched upward by a vertically-oriented spring, with spring constant k = 1.0 k N/m, that is compressed by an amount y = 21 cm. (a) How high does the ball rise above the launcher? (b) On the way down, at what speed does the ball strike a player’s glove which is held at the same height as the launcher?

+ x

+ yh max

Launcher

Ball

Let the spring release the ball at y = 0. Then, h max is some value of + y. Now, all of the potential energy stored in the compressed spring, Uspring, is converted to kinetic energy, K, just as the ball loses contact with the spring at y = 0.Here, at y = 0, ETotal = K + U and U = 0 because h = 0. As the ball rises, h > 0 so K begins to shift to U until, at the top,

K at the top is 0 because vTop = 0. On the way down, this process reverses which means vBottom on the way up equals vBottom on the way down. So, the total energy in the system equals the work expended to compress the spring. Stated mathematically:

TE K U 0

We are not interested in the velocity at the bottom, so

(a)

21( )

2k y m g h

Spring Bottom TopU K U

Spring TopU U

21( )

2k y

hm g

3 2

2

11.0 [10 ] (0.21 )

2

(0.180 ) (9.80 )

Nm

mm

kgs

12h m

(b) Now, let us find the velocity at the bottom. Earlier, we determined

and now we are not interested in UTop, so

Spring Bottom TopU K U

Spring BottomU K

2 21 1( )

2 2 Bottomk y m v

2( )Bottom

k yv

m

3 21.0 [10 ] (0.21 )

(0.180 )

Nm

mkg

16m

s