Many kinematics problems are given to students in high school and college physics courses which involve the five following measurable quantities of a body in motion:
a
{\displaystyle a\,}
acceleration (most often assumed to be constant)
t
{\displaystyle t\,}
time elapsed
v
f
{\displaystyle v_{f}\,}
velocity (this variable may also be represented by just
v
{\displaystyle v\,}
v
i
{\displaystyle v_{i}\,}
u
{\displaystyle u\,}
v
0
{\displaystyle v_{0}\,}
x
{\displaystyle x\,}
d
{\displaystyle d\,}
s
{\displaystyle s\,}
If one knows any three of the above quantities for a given situation, then one may solve for any of the other two. The equations below demonstrate this property.
Solving for (constant) acceleration
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Time is unknown
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a
=
v
f
2
−
v
i
2
2
x
{\displaystyle a={\dfrac {v_{f}^{2}-v_{i}^{2}}{2x}}\,}
Final velocity is unknown
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a
=
2
x
−
v
i
t
t
2
{\displaystyle a=2{\dfrac {x-v_{i}t}{t^{2}}}\,}
Initial velocity is unknown
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a
=
2
v
f
t
−
x
t
2
{\displaystyle a=2{\dfrac {v_{f}t-x}{t^{2}}}\,}
Distance is unknown
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a
=
v
f
−
v
i
t
{\displaystyle a={\dfrac {v_{f}-v_{i}}{t}}\,}
Solving for time
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Acceleration is unknown (but constant)
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t
=
2
x
v
f
+
v
i
{\displaystyle t={\dfrac {2x}{v_{f}+v_{i}}}\,}
Final velocity is unknown
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t
=
−
v
i
+
v
i
2
+
2
a
x
a
{\displaystyle t={\dfrac {-v_{i}+{\sqrt {v_{i}^{2}+2ax}}}{a}}\,}
Initial velocity is unknown
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t
=
−
v
f
+
v
f
2
−
2
a
x
−
a
{\displaystyle t={\dfrac {-v_{f}+{\sqrt {v_{f}^{2}-2ax}}}{-a}}\,}
Distance is unknown
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t
=
v
f
−
v
i
a
{\displaystyle t={\dfrac {v_{f}-v_{i}}{a}}\,}
Solving for final velocity
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Acceleration is unknown (but constant)
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v
f
=
2
x
t
−
v
i
{\displaystyle v_{f}={\dfrac {2x}{t}}-v_{i}\,}
Time is unknown
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v
f
=
v
i
2
+
2
a
x
{\displaystyle v_{f}={\sqrt {v_{i}^{2}+2ax}}\,}
Initial velocity is unknown
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v
f
=
2
x
+
a
t
2
2
t
{\displaystyle v_{f}={\dfrac {2x+at^{2}}{2t}}\,}
Distance is unknown
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v
f
=
v
i
+
a
t
{\displaystyle v_{f}=v_{i}+at\,}
Solving for initial velocity
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Acceleration is unknown (but constant)
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v
i
=
2
x
t
−
v
f
{\displaystyle v_{i}={\dfrac {2x}{t}}-v_{f}\,}
Time is unknown
edit
v
i
=
v
f
2
−
2
a
x
{\displaystyle v_{i}={\sqrt {v_{f}^{2}-2ax}}\,}
Final velocity is unknown
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v
i
=
2
x
−
a
t
2
2
t
{\displaystyle v_{i}={\dfrac {2x-at^{2}}{2t}}\,}
Distance is unknown
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v
i
=
v
f
−
a
t
{\displaystyle v_{i}=v_{f}-at\,}
Solving for distance
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Acceleration is unknown (but constant)
edit
x
=
v
f
+
v
i
2
t
{\displaystyle x={\dfrac {v_{f}+v_{i}}{2}}t\,}
Time is unknown
edit
x
=
v
f
2
−
v
i
2
2
a
{\displaystyle x={\dfrac {v_{f}^{2}-v_{i}^{2}}{2a}}\,}
Final velocity is unknown
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x
=
v
i
t
+
1
2
a
t
2
{\displaystyle x=v_{i}t+{\tfrac {1}{2}}at^{2}\,}
Initial velocity is unknown
edit
x
=
v
f
t
−
1
2
a
t
2
{\displaystyle x=v_{f}t-{\tfrac {1}{2}}at^{2}\,}
