Physics 105A
28 June 2016
Manuel Calderón de la Barca Sánchez
Sec 3.3: Newton’s Laws as diff. eqns.
 F=ma as a differential equation
mx = F(t, x,v)
– Force depends only on t (oscillatory driving forces)
 x=
dv
dt
– Force depends only on x (mass-spring system, simple harmonic
motion)
 x=v
dv
dx
– Force depends only on v (drag forces, turbulence)
 x = dv

Examples
dt
– Review: Falling under constant Gravitational Force
– F(v) case: Falling under constant Gravitational Force + air drag
28 June 2016
MCBS
Motion under constant g field.
 A particle of mass m is subject to a constant force F=-mg
directed down. The particle starts at rest at height h.
Solve for y(t) in two ways:
a = dv/dt
a = vdv/dx
28 June 2016
MCBS
Including air drag to a falling ball
 A physics prof. is dropped from rest at height h.
 Assume that the drag force from the air takes the form
Fd = -b v = -mav
 Solve the equations of motion to obtain v(t) and y(t).
28 June 2016
MCBS
Projectiles: Throwing a ball

For a given initial speed, at
which inclination angle should a
ball be thrown so that it travels
the max. horizontal distance by
the time it returns to the
ground?
Assume ground is horizontal, ball
is released at ground level.

What is the optimal angle if the
ground is sloped at an angle b
(where -90<b<90 degrees)?
28 June 2016
MCBS