SME
Meteorology
Clouds and precipitation
Use this for “A cloud droplet’s radius is 10 μm and a raindrop’s is 1 mm. How many droplets make the drop, and how long does a droplet take to fall 1 km?”
Diagram
10 = 1 × 10¹
10¹ ≤ 10 < 10²
10¹ = 10
10² = 100
This is an example. Type your own values to start a new problem.
r
Droplet radius
example · μm
R
Raindrop radius
example · mm
N
Droplets per drop
calculated
v
Droplet fall speed
calculated · cm/s
t
Time to fall 1 km
calculated
Formulas
N
=
(1,000 ×
R
r
)
3
1,000,000
=
(1,000 ×
1
10
)
3
v
=
1.19 ×
10
6
×
(
r
×
10
-4
)
2
1.19
=
1.19 ×
10
6
×
(10 ×
10
-4
)
2
t
=
100,000
v
84,033.6134
=
100,000
1.19
Formulas are worked in s. The step-by-step shows the conversions.
Assumptions
•
Stokes’ law, v = 1.19 × 10⁶ × r² with r in cm, holds for droplets under about 40 μm.
•
Still air: real updrafts hold droplets up, so alone they never fall out of the cloud.
•
Both are spheres, so the count is the cube of the radius ratio; 1 mm is 1,000 μm.
•
1 km is 100,000 cm, and a droplet reaches its fall speed almost at once.
Step-by-step
WE KNOW
r = 10 μm, R = 1 mm
FIND
droplets per drop (N), droplet fall speed (v), time to fall 1 km (t)
Step 1 · Find droplets per drop (N)
Use N = (1,000 × R ÷ r)³
Volume grows as the cube of the radius, so cube how many droplet radii fit across the drop’s.
N
=
(1,000 ×
1
10
)
3
N
=
100
3
N
=
1,000,000
Step 2 · Find droplet fall speed (v)
Use v = 1.19 × 10⁶ × (r × 10⁻⁴)²
Stokes:
drag
grows
with
speed
until
it
balances
the
weight;
r
×
10
-4
is
the
radius
in
cm.
v
=
1.19 ×
10
6
×
(10 ×
10
-4
)
2
v
=
(1.19
×
10
6
)
×
10
-6
v
=
1.19
cm/s
Step 3 · Find time to fall 1 km (t)
Use t = 100,000 ÷ v
1 km is 100,000 cm; divide it by the speed in cm/s.
t
=
100,000
1.19
t
=
84,033.6134
s
Step 4 · Convert the answers
Change each answer back to the unit you chose.
t = 84,033.6134 s = 23.3427 h (1 h = 3600 s)
Check
Put the numbers back in. Both sides should match.
1,000,000
=
(1,000 ×
1
10
)
3
✓
1.19
=
1.19 ×
10
6
×
(10 ×
10
-4
)
2
✓
84,033.61
=
100,000
1.19
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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