1D Kinematics

Section 3.1 1D Kinematics

Kinematics analyzes the positions and motions of objects as a function of time, without regard to the causes of motion. It involves the relationships between displacement (\(\vec{s}\)), velocity (\(\vec{v}\)), acceleration (\(\vec{a}\)), and time (t). The motion of object along one dimension has been analyzed in one dimensional (1D) kinematics.

Definition 3.1.2.

Displacement: It is the shortest distance between the initial position and the final position of an object. It has a particular direction and hence it is a vector quantity. In Figure 3.1.1.(a) an object is at rest because the position of the object is not changing with the time. In Figure 3.1.1.(b) object is moving along the curve path [\(x=x(t)\)] and hence the displacement after time [\(\Delta t = t_{f}-t_{i}\)] is given by [\(\,ds=x_{f}-x_{i}\)].

Definition 3.1.3.

Velocity: It is a vector expression of the speed traveled by a body in an specific direction of its motion. It is defined as a displacement over time or a rate of change of position. If the position of a particle at any time t is defined as \(x = x(t)\text{,}\) then its average velocity is given by

\begin{equation} v_{av}=\frac{x(t_{f})-x(t_{i})}{t_{f}-t_{i}} = \frac{x_{f}-x_{i} }{t_{f}-t_{i} } =\frac{\Delta s}{\Delta t}=\frac{\Delta x}{\Delta t}\tag{3.1.1} \end{equation}

where \(x_{f}, x_{i},t_{f},\) and \(t_{i}\) are final position, initial position, final time, and initial time, respectively. Average velocity of a body during some period can be determined by drawing the slope of the line passing through the points p and q on the curve. The average velocity can also be calculated by using the formula

\begin{equation*} v_{av} = \frac{v_{f}+v_{i} }{2 } \end{equation*}

if \(v_{i}\) and \(v_{f}\text{,}\) the initial and final velocities of a body are known and the motion of a body happen to be in a straight line with a constant acceleration.

Definition 3.1.4.

Instantaneous Velocity: It is a velocity of a moving object at a particular instant of time or at a particular position of its motion.

\begin{equation*} v_{inst} = \lim\limits_{\Delta t \to 0}\frac{\Delta s}{\Delta t} \end{equation*}

\begin{equation*} \text{or,}\quad v = \frac{\,ds}{\,dt} \end{equation*}

\begin{equation} \left[\text{or,}\quad \vec{v} = \frac{\,d\vec{s}}{\,dt}=\frac{\,d\vec{x}}{\,dt}\right] \tag{3.1.2} \end{equation}

The speedometer needle in a moving car gives instantaneous velocity of the car. Theoretically, it should be measured in the shortest time slice possible (\(\Delta t \to 0\)). Instantaneous velocity at a moment is the slope of tangent line drawn at a particular point on the curve as shown in Figure 3.1.1.(c).

Definition 3.1.5.

Acceleration: It is defined as a rate of change of velocity. It is a vector quantity. If the velocity of a particle at any time t is defined as \(v = v(t)\text{,}\) then its average acceleration is given by

\begin{equation} a_{av}=\frac{v(t_{f})-v(t_{i})}{t_{f}-t_{i}} = \frac{v_{f}-v_{i} }{t_{f}-t_{i} } =\frac{\Delta v}{\Delta t}\tag{3.1.3} \end{equation}

where \(v_{f}, v_{i},t_{f}, \) and \(t_{i}\) are final velocity, initial velocity, final time, and initial time, respectively Figure 3.1.1.(d). If velocity is decreasing with time then acceleration is negative or also called deceleration or retardation.

Definition 3.1.6.

Instantaneous Acceleration: It is an acceleration of a moving object at a particular instant of time. See Figure 3.1.1.(e)

\begin{equation*} a_{inst} = \lim\limits_{\Delta t \to 0}\frac{\Delta v}{\Delta t} \end{equation*}

\begin{equation*} \text{or,}\quad a = \frac{\,dv}{\,dt} \end{equation*}

\begin{equation} \left[\text{or,}\quad \vec{a} = \frac{\vec{dv}}{\,dt}=\frac{\,d^{2}\vec{x}}{dt^{2}}\right]\tag{3.1.4} \end{equation}

Acceleration produced by the pull of the earth (or the earth gravitational force) is called acceleration due to gravity, which is given by \(g = 9.8 m/s^{2}= 32 ft/s^{2}= 980 cm/s^{2}\text{.}\)

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