OCR A Level Mathematics A
Mechanics – Complete Notes
By Rujul Nayak (University of Cambridge O er-Holder)
Section 1: Kinematics 2
Section 2: Forces 6
ff
, OCR A Level Mathematics A Notes by Rujul Nayak
Section 1: Kinematics
SUVAT Equations
Kinematics is the branch of mechanics considering displacement, velocity, and acceleration as
functions of time, without considering the forces that cause the motion.
There are ve fundamental variables in kinematics:
• s, displacement (measured in metres, m)
• u, initial velocity (measured in metres per second, ms−1)
• v, nal velocity (measured in metres per second, ms−1)
• a, acceleration (measured in metres per second per second, ms−2)
• t, time (measured in seconds, s)
The acronym SUVAT is used to refer to these ve variables.
They are linked through di erentials: acceleration is the rate of change of velocity, which is the
rate of change of displacement.
dv ds
Therefore we know that, in any case, a = and v = .
dt dt
There are also ve SUVAT equations, which only hold when acceleration is constant. A graphical
method was used to derive them in the Physics notes (Section 4 “Mechanics”, Page 2-3), but we
will use calculus to derive some of them here:
Starting with acceleration, we can integrate it with respect to time to get the velocity, v. However,
we could also integrate normally to get at + c. Note that this constant c is the initial value of the
velocity when t = 0, therefore c = u. Thus we know that v = u + at, our rst SUVAT equation.
Now, integrating this equation again, we get displacement, s , on the left hand side, and
1 2
ut + at + c on the right hand side (however the initial displacement c will always be 0 ).
2
1
Therefore s = ut + at 2, our second SUVAT equation.
2
We could also rearrange the rst equation to get u = v − at , then integrate both sides to get
1 2
s = vt − at , our third SUVAT equation.
2
v−u
We can then substitute a = (from the rst equation) into the second one, resulting in
t
v−u 2 1
s = ut + t , which simpli es to s = (u + v) t, our fourth SUVAT equation.
2t 2
v−u
Finally, substituting t = (from the rst equation) into the third one gives
a
(v + u)(v − u)
s= , or v 2 = u 2 + 2a s, our fth SUVAT equation.
2a
Page 2 of 9
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Mechanics – Complete Notes
By Rujul Nayak (University of Cambridge O er-Holder)
Section 1: Kinematics 2
Section 2: Forces 6
ff
, OCR A Level Mathematics A Notes by Rujul Nayak
Section 1: Kinematics
SUVAT Equations
Kinematics is the branch of mechanics considering displacement, velocity, and acceleration as
functions of time, without considering the forces that cause the motion.
There are ve fundamental variables in kinematics:
• s, displacement (measured in metres, m)
• u, initial velocity (measured in metres per second, ms−1)
• v, nal velocity (measured in metres per second, ms−1)
• a, acceleration (measured in metres per second per second, ms−2)
• t, time (measured in seconds, s)
The acronym SUVAT is used to refer to these ve variables.
They are linked through di erentials: acceleration is the rate of change of velocity, which is the
rate of change of displacement.
dv ds
Therefore we know that, in any case, a = and v = .
dt dt
There are also ve SUVAT equations, which only hold when acceleration is constant. A graphical
method was used to derive them in the Physics notes (Section 4 “Mechanics”, Page 2-3), but we
will use calculus to derive some of them here:
Starting with acceleration, we can integrate it with respect to time to get the velocity, v. However,
we could also integrate normally to get at + c. Note that this constant c is the initial value of the
velocity when t = 0, therefore c = u. Thus we know that v = u + at, our rst SUVAT equation.
Now, integrating this equation again, we get displacement, s , on the left hand side, and
1 2
ut + at + c on the right hand side (however the initial displacement c will always be 0 ).
2
1
Therefore s = ut + at 2, our second SUVAT equation.
2
We could also rearrange the rst equation to get u = v − at , then integrate both sides to get
1 2
s = vt − at , our third SUVAT equation.
2
v−u
We can then substitute a = (from the rst equation) into the second one, resulting in
t
v−u 2 1
s = ut + t , which simpli es to s = (u + v) t, our fourth SUVAT equation.
2t 2
v−u
Finally, substituting t = (from the rst equation) into the third one gives
a
(v + u)(v − u)
s= , or v 2 = u 2 + 2a s, our fth SUVAT equation.
2a
Page 2 of 9
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