Motion - Notes
Physics - Notes
Motion
Motion is defined as the change in position of an object over time. It can be observed in many things around us, including people walking, water flowing, and celestial bodies moving.
Uniform Motion
Uniform motion is the state where an object covers equal distances in equal intervals of time, maintaining a constant speed and direction along a straight path. In this type of motion, the object's velocity remains constant, resulting in zero acceleration.
Non Uniform Motion
Non-uniform motion, also known as variable motion, occurs when an object's speed or direction, or both, change over time. This means the object covers unequal distances in equal intervals of time.
Quantity
A quantity is an attribute of a phenomenon, body, or substance that can be quantitatively described by numbers and units using measurement.
Scalar Quantity
A scalar quantity has only a magnitude (a numerical value and unit) and no direction, such as mass, time, or speed.
Vector Quantity
A vector quantity has both a magnitude and a specific direction to be fully described, like force, velocity, or displacement.
Distance & Displacement
Distance
Distance is the total length of the path an object travels, regardless of direction. It's a scalar quantity, which means it has magnitude but no direction
Displacement
Displacement is the change in position of an object from its starting point to its ending point. It is a vector quantity, meaning it has both magnitude and direction.
Speed & Velocity
Speed
Speed is defined as the rate at which an object's distance changes over time. It is a scalar quantity that measures how fast an object is moving.
Velocity
Velocity is defined as the rate at which an object changes its position over time in a specific direction. It's a vector quantity, which means it has both magnitude and direction. The SI unit of velocity is meters per second (m/s).
Acceleration & Retardation
Acceleration
Acceleration is defined as the rate of change of velocity, meaning it measures how quickly an object's velocity changes over time. The SI unit of acceleration is meters per second \(({m}/{s^2})\). It is a vector quantity. Acceleration \(\color{blue}{a}\) can be calculated using the formula \[\color{blue}{a=\frac{(v-u)}{t}}\] where \(\color{blue}{v}\) is the final velocity, \(\color{blue}{u}\) is the initial velocity, and \(\color{blue}{t}\) is the time taken for the change.
Retardation
Retardation is the negative acceleration of an object, meaning its velocity decreases over time. It's the rate at which an object's speed slows down. The SI unit for retardation is the same as acceleration: meters per second squared \(({m}/{s^2})\).
EQUATIONS OF MOTION
When the body is moving along a straight line with uniform acceleration, a relation can be established between the velocity of the body, the acceleration of the body and the distance travelled by the body in a specific time by a set of equations. These equations are called equations of motion. The three equations are:
- First Equation of motion : \(\color{blue}{v = u + at}\)
- Second Equation of motion : \(\color{blue}{s = ut + \frac{1}{2}at^2}\)
- Third Equation of motion : \(\color{blue}{v^2 - u^2 = 2as}\)
FIRST EQUATION OF MOTION
The first equation of motion, also known as the velocity-time relation, describes the relationship between final velocity, initial velocity, acceleration, and time for an object moving with uniform acceleration. It states that the final velocity \(\color{blue}{v}\) of an object is equal to the sum of its initial velocity \(\color{blue}{u}\) and the product of its acceleration \(\color{blue}{a}\) and the time \(\color{blue}{t}\) it has been accelerating.
\[\color{blue}{v = u + at}\]Consider a body having initial velocity \(\color{blue}{u}\). Suppose it is subjected to a uniform acceleration \(\color{blue}{a}\) so that after time \(\color{blue}{t}\) its final velocity becomes \(\color{blue}{v}\). Now we know,
Acceleration = change in velocity/time
This equation is known as the first equation of motion.
SECOND EQUATION OF MOTION
The second equation of motion describes the relationship between the position (displacement), initial velocity, time, and constant acceleration of an object moving in a straight line. It is also known as the position-time relation.
FormulaThe formula for the second equation of motion is given by:
\[\color{blue}{s=ut+\frac{1}{2}at^{2}}\]Derivation
Suppose a body has an initial velocity \(\color{blue}{u}\) and uniform acceleration \(\color{blue}{a}\) for time \(\color{blue}{t}\) so that its final velocity becomes \(\color{blue}{v}\). The distance travelled by moving body in time \(\color{blue}{t}\) is \(\color{blue}{s}\) then the average velocity\((v_{av})\) = \(\color{blue}{\frac{(v + u)}{2}}\).
Distance traveled\((s)\) = Average velocity (\(v_{av}) \times t \)
\[ \begin{align} s &= v_{av} \times t \nonumber \\\\ \text{but } v_{av} &= \frac{u+v}{2} \nonumber \\\\ &= \frac{u+v}{2} \times t \tag{i} \end{align} \]By first equation of motion, we can write
\[v=u+at\tag{ii}\]putting value of \(v\) from eqn \((ii)\) to eqn\((i)\)
\[ \require{cancel}\begin{aligned} s&=\frac{u + \color{orange}{(u+at)}}{2}\times t\\&=\left(\frac{2u+at}{2}\right)\times t\\&=\frac{2ut}{2} +\frac{at^2}{2}\\&=\frac{\cancel{2}ut}{\cancel2} + \frac{at^2}{2}\\&=ut + \frac{1}{2}at^2\\\Rightarrow s&=ut + \frac{1}{2}at^2 \end{aligned} \] \[ \begin{array}{|l|} \hline\color{blue}{s=ut + \frac{1}{2}at^2}\\\hline \end{array} \]THIRD EQUATION OF MOTION
The third equation of motion describes the relationship between the final velocity, initial velocity, acceleration, and displacement of an object moving with uniform acceleration. It is expressed as:
\[v^{2}=u^{2}+2as\]Where:
- \(v\) represents the final velocity.
- \(u\) represents the initial velocity.
- \(a\) represents the uniform acceleration.
- \(s\) represents the displacement.
Derivation
From the Second Law of Motion, Distance can be given as
\[s=\scriptsize ut+\frac{1}{2}at^2\tag{i}\]and from the First Law of Motion, we can write
\[\scriptsize(v=u+at)\Rightarrow t=\frac{(v-u)}{a}\tag{ii}\]
Putting value of (\(t\)) from eqn (ii) to eqn (i)
Graphical Representation of Motion
Graphs give a pictorial representation of motion. They make it easier to understand relationships between physical quantities. In motion, we usually use:
- Uniform Motion (constant speed):
- Straight line (positive slope).
- Slope represents Speed
- Steeper slope\(\Rightarrow\) higher speed
- Non-Uniform Motion:
- Curved line
- Slope changes at different points
- Object at Rest:
- Horizontal line parallel to x-axis.
- Distance does not change with time.
Distance-Time Graph
Velocity-Time Graph
- Uniform Velocity (no acceleration):
- Straight line parallel to the time axis.
- Displacement = Area under graph
- Uniform Acceleration:
- Straight line sloping upwards.
- Slope = Acceleration.
- Area under graph = Displacement.
- Uniform Retardation:
- Straight line sloping downwards.
- Object slowing down.
- Non-uniform Acceleration:
- Curved line.
- Important Uses of Graphs
- Slope of Distance–Time graph = Speed
\[ \text{Speed} = \frac{\Delta d}{\Delta t} \]
- Slope of Velocity–Time graph = Acceleration
\[ a = \frac{\Delta v}{\Delta t} \]
- Area under Velocity–Time graph = Displacement
- Rectangle area \(\Rightarrow\) uniform velocity
- Triangle + rectangle \(\Rightarrow\) uniformly accelerated motion
Uniform Circular Motion
Uniform circular motion describes an object moving along a perfectly circular path at a constant speed. The tangential velocity vector is always tangent to the circumference at each point. Tangential velocity is a vector aligned with tangential speed, maintaining a constant magnitude equal to the tangential speed of the motion. The formula for tangential speed (v) in uniform circular motion is given by:
\[\begin{array}{|l|}\hline \color{blue}{v= \frac{2\pi r}{T}}\\\hline\end{array}\]- where:
- \(v\) is the tangential velocity
- \(r\) is the radius
- \(T\) is the period (time required to complete one full circle)
Centripetal Acceleraion
Centripetal acceleration is the acceleration that causes an object to move in a circular
path. It's
always directed towards the centre of the circle, and its magnitude is determined by the object's
speed
and the radius of the circular path. Without centripetal acceleration, an object would move in a
straight line due to inertia.
Centripetal acceleration (a) in uniform circular motion is determined by the formula:
\[\begin{array}{|l|}\hline \color{blue}{a= \frac{v^2}{r}}\\\hline\end{array}\]
where: \(v\) is the tangential velocity, \(r\) is the radius.
Important Points to Remember
- Motion is a change of position; it can be described in terms of the distance moved or the displacement.
- The motion of an object could be uniform or non-uniform depending on whether its velocity is constant or changing.
- The speed of an object is the distance covered per unit time, and velocity is the displacement per unit time.
- The acceleration of an object is the change in velocity per unit time.
- Uniform and non-uniform motions of objects can be shown through graphs.
- The motion of an object moving at uniform acceleration can be described with the help of the
following equations, namely
- \(v = u + at\)
- \(s = ut + \frac{1}{2} at^2\)
- \(2as = v^2 – u^2\)
where \(u\) is the initial velocity of the object, which moves with uniform acceleration \(a\) for time \(t,~v\) is its final velocity and \(s\) is the distance it travelled in time \(t\). - If an object moves in a circular path with uniform speed, its motion is called uniform circular motion.
