Energy – The capacity to do work
1 Joule = 1 Newton-meter =1 kg-m2/s2
K = ½ mv2
U = mgy
Law of Conservation of Energy
½ mv21 + mgy1 = ½ mv22 + mgy2
Mr. Fredericks throws a 20.0 g pebble
straight up with a speed of 25 m/s.
What an afternoon of fun!
a. Calculate the maximum height of the
pebble. (32 m)
b. Calculate the speed at which it returns to
Mr. Fredericks’ hands.
Mr. Fredericks (100 kg)
jumps from a 3 meter
cliff. Calculate his
velocity when he is:
a) 2 m above the ground
b) 1 m above the ground
c) Just before he hits (0 m)
Assume the roller coaster car has a mass of
1000 kg. Calculate:
a) U gained from A B (9.8 x 104 J)
b) Total U at point B (2.45 x 105 J)
c) Speed at C. (22.1 m/s)
d) At what height will it have half the speed as
at C? ()
A child runs at 2.0 m/s and jumps onto her
sled. The sled is at the top of a 5.0 m hill.
a. Calculate her speed at the bottom.
Ballistic Pendulum
Used to determine
velocities
Perfectly Inelastic
collisions (KE
converts to PE)
Perfectly Inelastic Collision: Ex 3
A 5.00 gram bullet is fired into a 1.00 kg block
of wood. The wood-bullet system rises 5
cm. Calculate the initial velocity of the
bullet.
m1v1 + m2v2 = m1v1’ + m2v2’
m1v1 + m2v2 = (m1 + m2)vf
0 +(0.005 kg)(vi) = (1.00 kg + 0.005 kg)(vf)
(0.005 kg)(vi) = (1.005 kg)(vf)
Moment of impact
We need a second equation (two unknowns)
½ mv21 + mgy1 = ½ mv22 + mgy2
½(1.005 kg)(vf2) +0 =0 + (1.005 kg)(9.8 m/s2)(0.05 m)
vf = 0.990 m/s
(0.005 kg)(vi) = (1.005 kg)(0.990 m/s)
vi = 199 m/s
A 10.0 g bullet is fired into a 1.20 kg ballistic
pendulum. The bob rises to an angle of
40o. The string on the pendulum is 150 cm.
a. Calculate the height that the pendulum
rose. (35.1 cm)
b. Calculate the initial speed of the bullet.
(320 ms/)
An 8.00 g bullet is fired at 350 m/s into a
ballistic pendulum bob of 2.00 kg.
Calculate how high the coupled bullet
and bob will rise.
Hooke’s Law for Springs
F = -kx
F = weight of an object
k = spring constant (N/m)
x = displacement when the object is placed
on the spring
A mass of 25 grams is placed on a
spring. The spring stretches 6.00 cm
from the equilibrium position.
a. Calculate the spring constant. (4.1 N/m)
b. Calculate how far the spring would
stretch if 100 grams were placed in the
spring. (23.9 cm)
A 2.00 kg block is attached to a toy train
that moves at 5.0 cm/s. The block is
attached by a spring with spring
constant of 50 N/m and the ms = 0.60.
a. Derive a formula that indicates at what
length the block will slip.
b. Calculate the length the spring will stretch
before slipping. (23.5 cm)
c. Calculate the time at which that will
occur. (4.7 s)
Springs
• U = ½ kx2
• Springs are a way to store potential energy
½ mv2 + mgy + ½ ky2 = ½ mv2 + mgy + ½ky2
A 10.0 g toy dart is compresses a spring 6.0
cm. The spring’s constant is 10.0 N/m.
What speed will the dart leave the gun?
(1.90 m/s)
A 2.60 kg ball is dropped. It falls 55.0 cm
before hitting a spring. It compresses the
spring 15.0 cm before coming to rest.
Calculate the spring constant.
55.0 cm
-15.0 cm
A 2.00 kg block is placed on top of a 2.00 m
spring with a spring constant of 50,000
N/m. The spring is cranked down 80.0 cm
and released. How high will the block go?
(1800 m)
A vertical spring has a spring constant of 450
N/m and is mounted on the floor. A 0.30 kg
block is dropped from rest and compresses
the spring by 2.50 cm. Calculate the height
from which the block was dropped. Use the
unstretched spring length as your zero point.
(2.28 cm)
A vertical spring has a spring constant of
895 N/m is compressed by 15.0 cm.
a) Calculate the upward speed can it give to
a 0.360 kg ball when released. (7.28 m/s)
b) Calculate how high above the top of the
uncompressed spring the ball will fly. (2.70 m)
A 1.0 kg block and a 2.0 kg block are
connected by a 2000 N/m spring.
a. If the spring is compressed by 10 cm,
calculate the velocity of each block as it
flies apart. (Hint: use momentum and
conservation of energy) (3.6 m/s, 1.8 m/s)
Elastic Collisions
• Both KE and momentum are conserved
m1v1i + m2v2i = m1v1f + m2v2f
v1i + v1f = v2i + v2f
Elastic Collisions: Example 1
A cue ball moving at 2.0 m/s strikes the red
ball. What is the speed of both balls after
the elastic collision if they have equal
mass?
Elastic Collisions: Example 2
Two pool balls of equal mass collide. One is
moving to the right at 20 cm/s, and the
other to the left at 30 cm/s. Calculate their
velocities after they collide elastically.
+20 cm/s
-30 cm/s
Elastic Collisions: Example 3
A proton of mass 1.01 amu moving at 3.60 X
104 m/s elastically collides head on with a
still Helium nucleus (4.00 amu). What are
the velocities of the particles after the
collision?
Elastic collisions: Example 4
A 238 U atom decays into two small
“daughter” nuclei. The lighter one is going
east at 1.50 X 107 m/s, and the heavier
one west at 2.56 X 105 m/s. Calculate the
mass of each fragment. Assume they
both add to 238
A 200.0 g steel ball hangs from a 1.00 m long
string. The ball is released from a 45o
angle and strikes a 500 g block (neglect
friction).
a. Calculate the distance the ball will drop before
hitting the block. (29.3 cm)
b. Calculate the speed at which the ball will hit
the block. (2.40 m/s)
c. Calculate the velocity of the ball and block
after the collision. Assume an elastic collision.
(-1.03 and +1.37 m/s)
d. Calculate the angle the ball swings back to
(19o)