IEMDaily - Video Lecture Notes

Example

Velocity-displacement graph

Example:
The velocity

(v)

- distance

(s)

graph for an airplane travelling on a straight runway is shown. Find the acceleration (in

m / s^{2}

) of the plane at

s = 50 m

.
Solution:

a = v \frac{d v}{d x} = 20 \times \frac{40}{100} = 8

Definition

Feasibility of a graph

A graph is feasible if it does not violate the definition of a quantity. Some guidelines to be followed to check feasibility of a graph are:
1. Distance and speed are greater than zero.
2. Distance-time plot is increasing in nature.
3. Displacement/distance of a graph cannot change suddenly.
4. Quantity cannot take two values at an instant of time.
5. Graph does not exist in third or fourth quadrant since time is taken positive in normal scenarios.

Example

Inferring motion plots from given plots

Following the relationship between various motion parameters plots can be inferred for other parameter.

Example

Problems on vectors

To find the position vector of a particle at

x = 2 m, y = 1 m

with reference to an origin, its graph is drawn. The vector is represented by an arrow as shown. Position vector is found as

\vec{r} = 2 \hat{i} + \hat{j}

Definition

Acceleration in Motion of Bodies

When a car starts from a rest and travels in a straight line at increasing velocity, it is accelerating in the direction of travel (i.e linear accelaration). If the car changes direction, there is an acceleration toward the new direction (i.e non-linear accelaration).If the speed of the car decreases, this is an acceleration in the opposite direction to the direction of travel (i.e. deceleration).

Formula

Relation between position vector and velocity vector

Let,
P and Q are the initial and final position of a particle in time

Δ t

.
Position Vector of P and Q are,

{\vec{r}}_{1} = x_{1} \hat{i} + y_{2} \hat{j} + z_{2} \hat{k}

and

{\vec{r}}_{2} = x_{2} \hat{i} + y_{2} \hat{j} + z_{2} \hat{k}

Average velocity vector

{\vec{v}}_{a v} = \frac{Δ r}{Δ t}

{\vec{v}}_{a v}

\frac{{\vec{r}}_{2} - {\vec{r}}_{1}}{Δ t}

{\vec{v}}_{a v}

\frac{d \vec{r}}{d t}

Shortcut

Relation between acceleration vector and velocity vector

Let,
P and Q are the initial and final position of particle having velocities

v_{1} a n d v_{2}

respectively.

{\vec{r}}_{1} = x_{1} \hat{i} + y_{2} \hat{j} + z_{2} \hat{k}

and

{\vec{r}}_{2} = x_{2} \hat{i} + y_{2} \hat{j} + z_{2} \hat{k}

Average velocity vector

{\vec{v}}_{a v} = \frac{Δ r}{Δ t}

The average acceleration is defined as the rate at which velocity changes. It is in direction of change in veocity

Δ \vec{v}

.
Average acceleration =

{\vec{a}}_{a v} = \frac{Δ \vec{v}}{Δ t}

Formula

First equation of motion

Let an object is moving with uniform acceleration.
u = initial velocity of object
v = final velocity of object
a = uniform acceleration
Let object reaches at point B after time (t) Now, from the graph
Slope = Acceleration (a)=