Basic concepts
Dimensions
➔ Basic dimensions is length, time and mass and all mechanical quantities (velocity,
acceleration, force etc) can be expressed using these
➔ Units determined by SI
● Length: m
● Time: s
● Mass: kg
➔ Measurement is a process of finding the ratio of quantity/unit
giga G 10^9
mega M 10^6
kilo k 10^3
centi c 10^-2
milli m 10^-3
micro μ 10^-6
nano n 10^-9
Vectors
Scalars: Quantities that have a magnitude and no direction e.g. mass
Vectors: Quantities that have both a magnitude and direction e.g. force
➔ Represented in bold while their magnitudes are written ordinarily
➔ Added by joining together lines head to tail to give the “resultant”
➔ Vectors can also be subtracted from each other, to make a positive vector negative
just swap the direction
➔ Magnitude can be found graphically or by using pythagoras theorem and
trigonometry where the components of the vector are on the x and y axis.
● Be careful about whether x and y are positive or negative
➔ A vector can be multiplied by a scalar to give a vector in the same direction but with n
times the magnitude
Rectilinear motion
Frame of reference: Defined as a coordinate system (x,y,z axes) that can be at rest, moving,
rotating etc.
➔ Inertial frame of reference: At rest or moving at a constant speed in a straight line
Uniform motion: Motion in a straight line in which the distance covered in any two equal time
intervals is the same (i.e. constant speed)
,Displacement: The distance moved away from a frame of reference. Both distance moved
and direction is important.
Δx = xf - xi (x is horizontal distance)
Speed: Distance covered per unit of time
Velocity: Displacement per unit time
➔ Average velocity: v =Δx/Δt
➔ Instantaneous velocity is the rise/run of the displacement-time graph at a particular
time
Average acceleration: a = Δv/Δt
Instantaneous acceleration is the slope of the velocity-time graph at a particular time
➔ When velocity is 0 doesn’t mean that acceleration is 0 because a slope will still exist
on a velocity-time graph
If an object has a constant acceleration to the final velocity we can use: Vf = Vi + at
Similarly, constant acceleration also means that the average velocity = average of initial and
final velocities: v = (Vf + Vi)/2
Thus the distance traveled is: Δx = vt = ((Vf + Vi)/2) x t
Constantly accelerated motion
Galileo’s findings
➔ Falling bodies have identical speeds irrespective of weight and the final speed
depends only on the height fallen
➔ Air resistance increases with speed until terminal velocity is reached
➔ In the absence of resistance a body continues to move in a straight line
➔ For naturally accelerated motion velocity is proportional to time and change in
displacement is proportional to t^2
➔ The vertical and horizontal components of a compound motion are independent
➔ The path of a projectile is a parabola
Acceleration is -9.8m/s
Kinematic equations:
d = vxt
Average velocity (v) = (vf+vi)/2
ΔV = a x Δt
Final velocity: vf = vi +at
Final displacement: xf = xi + vt
Final displacement: xf = xi + (vi x t) + (½ a t^2)
Final velocity: vf^2 = vi^2 + 2aΔx
Two dimensional motion
,Projectile motion: Any body or mass projected above the surface of the earth that is
unpowered. Only force acting is gravity
➔ Horizontal and vertical components of projectile motion are independent of one
another
➔ In the horizontal direction there is no force and the motion is constant while in the
vertical direction the motion is uniformly accelerated by gravity (9.8) so will decrease
by 9.8 per second going up and increase by 9.8 per second coming down.
Vectors of motion
Vertical component is always ..sinθ
➔ Velocity
➔ Acceleration
➔ Position (y)
Horizontal component is always ...cosθ
➔ Velocity
➔ Acceleration
➔ Position (x)
T = 2vi sinθ/ g
Y max = (vi sinθ)^g
X max = vi^2 sin(2θ) / g
θ can be θi or 90-θi
Circular motion
Direction of motion for a body of mass changes when acceleration is in a different direction
to velocity.
➔ If acceleration has a constant magnitude and direction is perpendicular to the velocity
vector then the motion will be circular with a radius of R
➔ A = v^2 / R
➔ F = mv^2/r
Relative motion
Vab : Where the first number represents what is being observed/measured and the second
number represents who is observing/measuring (aka relative)
One dimension:
Vab = Vad + Vdb
➔ Must ensure that the values cancel algebraically, if given in the wrong order then
keep the magnitude constant but make the direction opposite
Two dimensions:
Vab = √ Vad^2 + Vdb^2
, Inertial reference frame: When we assume that we are viewing movement from a stationary
position
➔ Newton’s 1st law: In the absence of any external forces, when viewed from an inertial
reference frame, an object maintains inertia (rest remains rest, motion remains
constant)
➔ Newtons 2nd law: When there is a net (resultant) force on a body it undergoes
acceleration proportional to the force and inversely proportional to its mass
➔ Newtons 3rd law: When one body (A) exerts a force on another (B) the second body
exerts an equal and opposite force on the first.
Equations
F = ma
W = mg
F = kΔx
Box on a slope - Rearrange to vector and solve
Normal force: A contact force that surfaces exert to prevent solid objects from passing
through each other. If two surfaces are not in contact, they can't exert a normal force on
each other.
➔ Acts perpendicular to the slope
Friction and acceleration act along a slope
Weight force acts vertically
Statics
Fab = -Fba
When there is no motion, the force of static friction balances the opposite force up until
motion occurs in which friction is known as the kinetic friction.
Ffr ≤ usN
Ffr = ukN
Uk = coefficient of kinetic friction
Us = coefficient of static friction
Dimensions
➔ Basic dimensions is length, time and mass and all mechanical quantities (velocity,
acceleration, force etc) can be expressed using these
➔ Units determined by SI
● Length: m
● Time: s
● Mass: kg
➔ Measurement is a process of finding the ratio of quantity/unit
giga G 10^9
mega M 10^6
kilo k 10^3
centi c 10^-2
milli m 10^-3
micro μ 10^-6
nano n 10^-9
Vectors
Scalars: Quantities that have a magnitude and no direction e.g. mass
Vectors: Quantities that have both a magnitude and direction e.g. force
➔ Represented in bold while their magnitudes are written ordinarily
➔ Added by joining together lines head to tail to give the “resultant”
➔ Vectors can also be subtracted from each other, to make a positive vector negative
just swap the direction
➔ Magnitude can be found graphically or by using pythagoras theorem and
trigonometry where the components of the vector are on the x and y axis.
● Be careful about whether x and y are positive or negative
➔ A vector can be multiplied by a scalar to give a vector in the same direction but with n
times the magnitude
Rectilinear motion
Frame of reference: Defined as a coordinate system (x,y,z axes) that can be at rest, moving,
rotating etc.
➔ Inertial frame of reference: At rest or moving at a constant speed in a straight line
Uniform motion: Motion in a straight line in which the distance covered in any two equal time
intervals is the same (i.e. constant speed)
,Displacement: The distance moved away from a frame of reference. Both distance moved
and direction is important.
Δx = xf - xi (x is horizontal distance)
Speed: Distance covered per unit of time
Velocity: Displacement per unit time
➔ Average velocity: v =Δx/Δt
➔ Instantaneous velocity is the rise/run of the displacement-time graph at a particular
time
Average acceleration: a = Δv/Δt
Instantaneous acceleration is the slope of the velocity-time graph at a particular time
➔ When velocity is 0 doesn’t mean that acceleration is 0 because a slope will still exist
on a velocity-time graph
If an object has a constant acceleration to the final velocity we can use: Vf = Vi + at
Similarly, constant acceleration also means that the average velocity = average of initial and
final velocities: v = (Vf + Vi)/2
Thus the distance traveled is: Δx = vt = ((Vf + Vi)/2) x t
Constantly accelerated motion
Galileo’s findings
➔ Falling bodies have identical speeds irrespective of weight and the final speed
depends only on the height fallen
➔ Air resistance increases with speed until terminal velocity is reached
➔ In the absence of resistance a body continues to move in a straight line
➔ For naturally accelerated motion velocity is proportional to time and change in
displacement is proportional to t^2
➔ The vertical and horizontal components of a compound motion are independent
➔ The path of a projectile is a parabola
Acceleration is -9.8m/s
Kinematic equations:
d = vxt
Average velocity (v) = (vf+vi)/2
ΔV = a x Δt
Final velocity: vf = vi +at
Final displacement: xf = xi + vt
Final displacement: xf = xi + (vi x t) + (½ a t^2)
Final velocity: vf^2 = vi^2 + 2aΔx
Two dimensional motion
,Projectile motion: Any body or mass projected above the surface of the earth that is
unpowered. Only force acting is gravity
➔ Horizontal and vertical components of projectile motion are independent of one
another
➔ In the horizontal direction there is no force and the motion is constant while in the
vertical direction the motion is uniformly accelerated by gravity (9.8) so will decrease
by 9.8 per second going up and increase by 9.8 per second coming down.
Vectors of motion
Vertical component is always ..sinθ
➔ Velocity
➔ Acceleration
➔ Position (y)
Horizontal component is always ...cosθ
➔ Velocity
➔ Acceleration
➔ Position (x)
T = 2vi sinθ/ g
Y max = (vi sinθ)^g
X max = vi^2 sin(2θ) / g
θ can be θi or 90-θi
Circular motion
Direction of motion for a body of mass changes when acceleration is in a different direction
to velocity.
➔ If acceleration has a constant magnitude and direction is perpendicular to the velocity
vector then the motion will be circular with a radius of R
➔ A = v^2 / R
➔ F = mv^2/r
Relative motion
Vab : Where the first number represents what is being observed/measured and the second
number represents who is observing/measuring (aka relative)
One dimension:
Vab = Vad + Vdb
➔ Must ensure that the values cancel algebraically, if given in the wrong order then
keep the magnitude constant but make the direction opposite
Two dimensions:
Vab = √ Vad^2 + Vdb^2
, Inertial reference frame: When we assume that we are viewing movement from a stationary
position
➔ Newton’s 1st law: In the absence of any external forces, when viewed from an inertial
reference frame, an object maintains inertia (rest remains rest, motion remains
constant)
➔ Newtons 2nd law: When there is a net (resultant) force on a body it undergoes
acceleration proportional to the force and inversely proportional to its mass
➔ Newtons 3rd law: When one body (A) exerts a force on another (B) the second body
exerts an equal and opposite force on the first.
Equations
F = ma
W = mg
F = kΔx
Box on a slope - Rearrange to vector and solve
Normal force: A contact force that surfaces exert to prevent solid objects from passing
through each other. If two surfaces are not in contact, they can't exert a normal force on
each other.
➔ Acts perpendicular to the slope
Friction and acceleration act along a slope
Weight force acts vertically
Statics
Fab = -Fba
When there is no motion, the force of static friction balances the opposite force up until
motion occurs in which friction is known as the kinetic friction.
Ffr ≤ usN
Ffr = ukN
Uk = coefficient of kinetic friction
Us = coefficient of static friction
