Free Fall. Apollo 15 Astronaut David Scott perform. Galileo's experiment on the
Moon. Feather drop. Free Fall. ➢ Motion under influence of gravity alone.
Review:
Do you remember the four “kinematic equations worth committing to memory” from the last lecture?
Free Fall
For a = constant in time (only!) Then:
Vxf (t) = vxi − at xf (t) = xi + vxi t − ½ at2
Also:
Q1
1 (v xi + v xf ) 2 v xf2 = v xi2 − 2 g ( x f − xi ) vx =
Basic
Derived
Apollo 15 Astronaut David Scott perform Galileo's experiment on the Moon Feather drop
Special Case:
Free Fall
ay=constant= −9.8m/s2
Motion under influence of gravity alone (no air resistance) Near the surface of the earth: a = -g = -9.8 m/s2 Acceleration is always down, regardless of the motion (moving up, moving down, instantaneously at rest) Independent of size, shape, composition, mass
Then:
vyf = vyi − gt yf = yi + vyi t − ½ gt2 Also:
1 (v yi + v yf ) 2 v yf2 = v yi2 − 2 g ( y f − yi ) vy =
Basic
Derived
Q2
1
Example: Drop a ball from rest at a height 1.50 m above the ground. (a) How long before it hits the floor? (b) How fast is it moving when it hits?
Kinematics & Calculus Let’s start with a(t) and integrate:
a=
Strategy: 1. Draw a picture. 2. Label the “knowns” and identify the “unknowns”. 3. Choose a coordinate system and positive direction 4. Choose appropriate formulae a. make sure they apply b. try simplest first 5. Work algebra neatly. 6. Practice!
Example: a=constant dv = a dt = a dt
v(t ) = at + C1 Initial condition: vi= v(0) at t = 0
vi = C1 So:
v(t ) = vi + at
dv dt
dv = a dt
dv = a dt
Similarly: Let’s start with v(t) and integrate:
v=
dx dt
dx = v dt
dx = v dt
2
and for a=constant:
v(t ) = vi + at
dx = v dt = (vi + at )dt
x(t ) = vi t + 12 at 2 + C2 Initial condition: xi= x(0) at t = 0
xi = C2 So:
x(t ) = xi + vi t + 12 at 2
Grahically: (a=constant) v(t) vf
vi
lope a =s
atf
½atf2
vitf tf
t
∆x = xf – x i= vitf + ½atf2 = AREA under the curve
Extra Example: Throw a ball up with initial speed vi. A) How high does it rise? B) How long does it take to get to the top? C) How fast is it moving when it comes back to its original height?