Motion in a Straight Line: Comprehensive NEET Physics Notes

1. Introduction to Motion

Motion is a fundamental concept in physics, observed in various forms such as walking, running, or the rotation of the Earth. In this chapter, we will focus on the motion of objects along a straight line, known as rectilinear motion, and introduce the concepts of velocity, acceleration, and relative velocity.

1.1 Instantaneous Velocity and Speed

Instantaneous velocity is defined as the limit of the average velocity as the time interval becomes infinitesimally small. Mathematically, it is expressed as:

$v = lim_{Δ t \to 0} \frac{Δ x}{Δ t} = \frac{d x}{d t}$

Instantaneous speed is the magnitude of instantaneous velocity. For example, a velocity of $+ 24.0 m / s$ and a velocity of $- 24.0 m / s$ both have an associated speed of 2 $24.0 m / s$ .

Did You Know?

The speedometer in a car measures the instantaneous speed of the vehicle.

Mnemonic:

To remember the difference between speed and velocity, think: "Speed is Scalar, Velocity is Vector."

1.2 Acceleration

Acceleration is the rate of change of velocity with respect to time. The average acceleration over a time interval is given by:

$a = \frac{v _{2} - v _{1}}{t _{2} - t _{1}}$

Instantaneous acceleration is defined as the limit of the average acceleration as the time interval approaches zero:

$a = lim_{Δ t \to 0} \frac{Δ v}{Δ t} = \frac{d v}{d t}$

Common Misconception:

Acceleration can be negative. This does not mean the object is slowing down; it means the acceleration is in the direction opposite to the velocity.

1.3 Kinematic Equations for Uniformly Accelerated Motion

For uniformly accelerated motion, we derive simple equations relating displacement ( $x$ t), time ( $t$ ), initial velocity $v_{0}$ , final velocity ( $v$ ), and acceleration ( $a$ ):

$v = v_{0} + a t$
$x = v_{0} t + \frac{1}{2} a t^{2}$
$v^{2} = v_{0}^{2} + 2 a x$

These equations are essential for solving problems involving constant acceleration.

NEET Problem-Solving Strategy:

When solving problems with kinematic equations, list all known values and choose the equation that includes the unknown variable you need to find.

Real-life Application:

These equations are used to design roller coasters, ensuring that the acceleration experienced by riders is within safe limits.

1.4 Relative Velocity

Relative velocity is the velocity of one object as observed from another moving object. If two objects have velocities $v_{A}$ and $v_{B}$ , the relative velocity of object A with respect to B is:

$v_{A B} = v_{A} - v_{B}$

NEET Tip:

In relative velocity problems, carefully choose the frame of reference to simplify calculations.

2. Quick Recap

Motion is the change in position with time.
Instantaneous velocity is the derivative of position with respect to time.
Acceleration is the rate of change of velocity with time.
Kinematic equations relate displacement, time, initial and final velocities, and acceleration.
Relative velocity is the velocity of one object as observed from another moving object.

3. Practice Questions

Question 1

A car accelerates uniformly from rest to a speed of 20 m/s in 5 seconds. Calculate the acceleration and the distance traveled.

Solution: Given: $v_{0} = 0$ , $v = 20 m / s$ , $t = 5 s$ Usin $v = v_{0} + a t$ :

$20 = 0 + a \cdot 5 ⟹ a = 4 m / s^{2}$

Using $x = v_{0} t + \frac{1}{2} a t^{2}$ :

$x = 0 \cdot 5 + \frac{1}{2} \cdot 4 \cdot 5^{2} = 50 m$

Question 2

A ball is thrown vertically upward with a speed of 15 m/s. How high will it rise?

Solution: Given: $v_{0} = 15 m / s$ , $v = 0$ (at the highest point) $a = - 9.8 m / s^{2}$ Using $v^{2} = v_{0}^{2} + 2 a x$ :

$0 = 1 5^{2} + 2 (- 9.8) x ⟹ x = \frac{225}{19.6} = 11.48 m$

Question 3

A train moving with a velocity of 60 km/h is brought to rest in 2 minutes. Calculate the retardation.

Solution: Given: $v_{0} = 60 km / h = 16.67 m / s$ v0=60km/h=16.67m/s, $v = 0$ , $t = 120 s$ Using $v = v_{0} + a t$ :

$0 = 16.67 + a \cdot 120 ⟹ a = - 0.139 m / s^{2}$

Question 4

Two cars A and B are moving in the same direction with velocities 40 m/s and 30 m/s, respectively. What is the velocity of A relative to B?

Solution:

$v_{A B} = v_{A} -$