SME
Dynamics · Problem type
Normal and tangential acceleration on a curve
Use this for “A car at 20 m/s speeds up at 3 m/s² on a curve of radius 100 m. Find its normal and total acceleration.”
Diagram
aₙ
=
v
2
ρ
=
20
2
100
=
4
m/
s
2
aₙ points toward the center of curvature.
a
=
√
aₜ
2
+
aₙ
2
=
√
3
2
+
4
2
a
=
5
m/
s
2
ρ is drawn the same length whatever its value: the arrows are to scale, the path is not.
This is an example. Type your own values to start a new problem.
aₙ (m/s²)
=
(
v (m/s)
)
2
ρ (m)
a (m/s²)
=
(
aₜ (m/s²)
)
2
+
(
aₙ (m/s²)
)
2
Formulas
aₙ
=
v
2
ρ
4
=
20
2
100
a
=
√
aₜ
2
+
aₙ
2
5
=
√
3
2
+
4
2
Assumptions
•
Path coordinates: t points along the path the way the particle moves, n toward the center of curvature.
•
aₜ = dv/dt changes the speed (− when slowing down); aₙ = v² ÷ ρ turns the velocity.
•
The two parts are at right angles, so a = √(aₜ² + aₙ²).
Step-by-step
WE KNOW
v = 20 m/s, ρ = 100 m, aₜ = 3 m/s²
FIND
normal acceleration (aₙ), acceleration (a)
Step 1 · Find normal acceleration (aₙ)
Use aₙ = v² ÷ ρ
The
normal
part
turns
the
path
toward
the
center
of
curvature:
v
2
ρ
.
aₙ
=
20
2
100
aₙ
=
400
100
aₙ
=
4
m/
s
2
Step 2 · Find acceleration (a)
Use a = √(aₜ² + aₙ²)
The two parts are at right angles, so they add like the legs of a right triangle.
a
=
√
3
2
+
4
2
a
=
√
9 + 16
a
=
√
25
a
=
5
m/
s
2
Check
Put the numbers back in. Both sides should match.
4
=
20
2
100
✓
5
=
√
3
2
+
4
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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