SME
Meteorology · Problem type
The updraft that holds up a hailstone
Use this for “How strong an updraft keeps a 4 cm hailstone aloft?”
Diagram
Drag = 0.2959 N at 90° from the horizontal: (0, 0.2959) N.
W = 0.2959 N at 90° from the horizontal: (0, −0.2959) N.
ΣFₓ = 0 N and ΣFᵧ = 0 N.
The forces balance: tip to tail, the polygon closes.
This is an example. Type your own values to start a new problem.
v (m/s)
=
8g
×
r (cm)
×
ρice (kg/m³)
3
×
×
ρₐᵢᵣ (kg/m³)
d
Hailstone diameter
example
W
Weight
worked out (not typed) · N
Formulas
r
=
d
2
0.02
=
0.04
2
v
=
√(8
×
9.81
×
r
×
ρice
3 × Cd × ρₐᵢᵣ
)
28.0143
=
√(8
×
9.81
×
0.02
×
900
3 × 0.6 × 1
)
W
=
(
4
3
)
×
π
×
r
3
×
ρice
×
9.81
0.2959
=
(
4
3
)
×
π
×
0.02
3
×
900
×
9.81
Formulas are worked in m. The step-by-step shows the conversions.
Assumptions
•
The stone is a sphere at its terminal speed: its weight (4 ÷ 3)πr³ρice g equals the drag ½Cdρₐᵢᵣ πr²v².
•
Cd is about 0.6 for hail, and g = 9.81 m/s².
•
An updraft faster than v carries the stone up; a slower one lets it fall.
Step-by-step
WE KNOW
d = 4 cm, ρice = 900 kg/m³, ρₐᵢᵣ = 1 kg/m³, Cd = 0.6
FIND
radius (r), fall speed (v), weight (W)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to m).
d = 4 cm = 0.04 m (1 m = 100 cm)
Step 2 · Find radius (r)
Use r = d ÷ 2
The radius is half the diameter.
r
=
0.04
2
r
=
0.02
m
Step 3 · Find fall speed (v)
Use v = √(8 × 9.81 × r × ρice ÷ (3 × Cd × ρₐᵢᵣ))
Set
the
weight
equal
to
the
drag;
πr²
cancels,
leaving
v
2
=
8g
rρice
3Cdρₐᵢᵣ
.
v
=
√(8
×
9.81
×
0.02
×
900
3 × 0.6 × 1
)
8 × 9.81 × 0.02 × 900 = 1,412.6
3 × 0.6 × 1 = 1.8
1,412.6
1.8
=
784.8
v
=
28.0143
m/s
Step 4 · Find weight (W)
Use W = (4 ÷ 3) × π × r³ × ρice × 9.81
The ball’s volume times the density of ice is its mass; times g, its weight, which the drag holds up.
W
=
(
4
3
)
×
π
×
0.02
3
×
900
×
9.81
W
=
4
3
×
(8.0000
×
10
-6
)
×
900
×
9.81
×
π
W
=
1.3333
×
(8.0000
×
10
-6
)
×
900
×
9.81
×
π
W
=
(1.0666
×
10
-5
)
×
900
×
9.81
×
π
W
=
0.0095994
×
9.81
×
π
W
=
0.09417
×
π
W
=
0.2959
N
Step 5 · Convert the answers
Change each answer back to the unit you chose.
r = 0.02 m = 2 cm (1 m = 100 cm)
Check
Put the numbers back in. Both sides should match.
0.02
=
0.04
2
✓
28.0143
=
√(8
×
9.81
×
0.02
×
900
3 × 0.6 × 1
)
✓
0.2959
=
(
4
3
)
×
π
×
0.02
3
×
900
×
9.81
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.