Motion in a Straight Line Class 11 Notes

CBSE board students must be aware of what is instantaneous speed. This is one of the key concepts in motion in a straight line class 11 notes. In simple terms, it measures how fast an object is moving at a specific moment in time. Instantaneous speed provides the magnitude of the velocity vector at a particular instant. This speed is the limit of average speed as the time interval approaches zero. If the position of the object is given by function x(t), then the instantaneous speed v(t) at time t will be the absolute value of the derivative of x(t) with respect to time:

$v \left(t\right) = \left| \frac{ d x \left(t\right) }{ d t } \right|$

Instantaneous velocity is a vector quantity that describes the rate at which an object moves at a specific moment in time. It considers both speed of objects and its direction. Unlike average velocity, which is calculated over a certain time interval, instantaneous velocity gives information about the motion of an object at an exact instant. Do remember that NEET exam and JEE Main entrance exam, will not be asking direct questions based on the definition.

In mathematical terms, instantaneous velocity indicates the limit of average velocity as time interval approaches zero. If the position of an object is given by function x(t), then, instantaneous velocit v(t) at time t is the derivative of x(t) with respect to time:

$v \left(t\right) = \frac{ d x \left(t\right) }{ d t }$

The magnitude of the instantaneous velocity vector is the instantaneous speed. Instantaneous velocity can tell how fast an object is moving at a specific moment, but it also indicates the direction it is heading. 

These equations describe the relationships between object's displacement, acceleration, velocity and time. These are applicable when objects move with constant acceleration. The IIT JAM exam, often asks questions based on these formulas. Do remember that these questions will be application-based.The four primary kinematic equations include the following:

1. Velocity as a Function of Time

v = u+at

Here:

• v is final velocity
• u is initial velocity a
• a is the constant acceleration 
• t is the time

2. Displacement as a Function of Time

$s = u t + \frac{1}{2} a t^2$

Here:

s is displacement.

3. Velocity as a Function of Displacement

$v^2 = u^2 + 2 a s$

4. Displacement as a Function of Average Velocity

$s = \left( \frac{ u + v }{2} \right) t$

Relative velocity is a concept that describes an object's velocity as observed from a particular frame of reference. It is the velocity of one object relative to another moving object. This velocity depends on the reference frame from which the observation is made. Say two objects, A and B, are moving with velocities $\stackrel{\to }{v_A} \text{ and } \stackrel{\to }{v_B}$ respectively, the relative velocity of object A in respect of object B will be $\stackrel{\to }{v_{ A B }} = \stackrel{\to }{v_A} - \stackrel{\to }{v_B}$ .

Acceleration describes the rate at which velocity of an object changes over time. This vector quantity involves a change in both direction and speed. In mathematical terms, acceleration is:

$\stackrel{\to }{a} = \frac{ d \stackrel{\to }{v} }{ d t }$

The SI unit of acceleration is meters per second squared $( m / s^2 )$ .

There are different types of acceleration, including the following:

• Linear acceleration is a type where the velocity of an object changes uniformly in a straight line.
• Angular acceleration indicates the rate of change of angular velocity of an object that moves along a circular path.
• Centripetal acceleration is experienced by those objects that move in a circular path. This acceleration is directed towards the center of a circle and it causes the change in direction of velocity vector.
• Tangential acceleration occurs when an object in circular motion changes its speed. This is tangential to the circular path and impacts the magnitude of velocity vector.
• Uniform acceleration happens when velocity of an object changes at a constant rate over time. In this case, the acceleration stays the same in magnitude and direction.

Let us take a look at some of the frequently asked questions related to motion in a straight line: