Displacement, Velocity, and Acceleration of a Particle in Two Dimensional Motion and Their Vector Representation
To study about displacement, velocity and acceleration of a particle in two dimensional motion we study the about the motion of the particle in XY axis of the reference frame. Assume that at any time interval t1 the position of the particle is A1 whose position vector is \(\overrightarrow{r_{1}}\) and at any time t2 the position of the particle is A2 whose position vector is \(\overrightarrow{r_{2}}\). Therefore, the vector representation of points A1 and A2 is following;
Where \(\hat{i}\) and \(\hat{j}\) are unit vectors in the direction of x and y axis representing directions respectively.
and \(\vec{r}_{2}=x_{2} \hat{i}+y_{2} \hat{j}\) …………… (2)
Again, because the particle is displaced from A1 to A2. Hence, according to the diagram if displacement is Δr then according to the triangle rule in the triangle OA1A2.
Therefore, the displacement equation would be given as equation (3) in two dimensional motion.
According to Vector Algebra the resultant is;
Δr = \(\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}\) ………………… (4)
This equation (3) is the result of displacement and the direction of displacement vector Δr would be according to the figure.
To calculate the average velocity of the particle according to the definition of average velocity;
Hence, average velocity would be given as equation (5) and the resultant of average velocity by
Vector Algebra would be;
v = \(\sqrt{\left(v_{x}\right)^{2}+\left(v_{y}\right)^{2}}\) ……………. (6)
given by equation (6) and the direction of it would be in the direction of \(\overrightarrow{\Delta r}\). The instantaneous velocity of this particle at any instant t by the definition of instantaneous velocity would be;
Hence, instantaneous velocity is given by the eqn. (7). Resultant of instantaneous velocity by Vector Algebra.
v = \(\sqrt{v_{x}^{2}+v_{y}^{2}}\) …………….. (8)
will be given by eqn. (3) and direction would be in the direction of the tangent at the point; which is the position of the particle at time t.
Suppose it makes an angle θ with the x-axis; then the direction would be as shown in the figure (9) and by the equation;
tan θ = \(\frac{v_{y}}{v_{x}}\)
or θ = \(\tan ^{-1}\left[\frac{v_{y}}{v_{x}}\right]\) ……………….. (9)
If the object is executing accelerated motion and in the time interval ∆t the change in velocity is ∆v, then by the definition of average acceleration, the acceleration of the particle will be;
This is the equation of average acceleration.
Result of this by Vector Algebra is given by
a = \(\sqrt{\left(a_{x}\right)^{2}+\left(a_{y}\right)^{2}}\) ………………. (11)
will be given by equation (11) and the direction of it would be in the direction of ∆v.
To calculate instantaneous acceleration by
Equation (12) is the vector equation of instantaneous acceleration; and its result;
a = \(\sqrt{\left(a_{x}\right)^{2}+\left(a_{y}\right)^{2}}\) …………… (13)
will be given by (13) and its direction would be in the direction of v. All the various equations above explain displacement, velocity and acceleration in two-dimensional motion.
