SME
Grade 11 Science · Problem type
Components of a velocity
Use this for “A ball leaves at 15 m/s, 40° above level. What are its horizontal and vertical velocities?”
Diagram
v: 15 m/s at 40°, so v = ⟨11.49, 9.64⟩.
Horizontal velocity: 11.5 m/s
Vertical velocity: 9.64 m/s
This is an example. Type your own values to start a new problem.
vₓ (m/s)
=
v (m/s)
×
cos(
θ
°)
vᵧ (m/s)
=
v (m/s)
×
sin(
θ
°)
Formulas
vₓ
=
v
×
cos(
θ
)
11.5 = 15 × cos(40°)
vᵧ
=
v
×
sin(
θ
)
9.64 = 15 × sin(40°)
v
=
√
vₓ² +
vᵧ
2
15
=
√
11.5
2
+
9.64
2
tan(
θ
)
=
vᵧ/vₓ
tan(40°) = 9.64/11.5
Assumptions
•
The angle is measured from the horizontal (east), counterclockwise.
•
The two components are the legs of a right triangle; the speed is its hypotenuse.
Step-by-step
WE KNOW
v = 15 m/s, θ = 40°
FIND
horizontal velocity (vₓ), vertical velocity (vᵧ)
Step 1 · Find horizontal velocity (vₓ)
Use vₓ = v × cos(θ)
The
horizontal
part
is
the
speed
times
cos
θ
.
vₓ = 15 × cos(40°)
vₓ = 15 × 0.766
vₓ = 11.5 m/s
Step 2 · Find vertical velocity (vᵧ)
Use vᵧ = v × sin(θ)
The
vertical
part
is
the
speed
times
sin
θ
.
vᵧ = 15 × sin(40°)
vᵧ = 15 × 0.6428
vᵧ = 9.64 m/s
Check
Put the numbers back in. Both sides should match.
11.5 ≈ 15 × cos(40°) ✓
9.64 ≈ 15 × sin(40°) ✓
15
≈
√
11.5
2
+
9.64
2
✓
tan(40°) = 9.6418/11.491 ✓
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