Kinematics Notes for NEET Exam
1. The Frame of Reference
Definition: A frame of reference is a coordinate system used to describe the position and motion of an object.
Types:
- Inertial Frame: A frame that is not accelerating (e.g., a stationary frame or one moving at constant velocity).
- Non-Inertial Frame: A frame that is accelerating (e.g., a rotating frame).
NEET PYQ 2020: Which of the following is an inertial frame of reference?
A) A rotating merry-go-round
B) A car moving at constant velocity
C) A freely falling elevator
D) A train accelerating on a track
Answer: B) A car moving at constant velocity
Explanation: An inertial frame is one that is not accelerating. A car moving at constant velocity is an inertial frame, while the other options involve acceleration.
NEET PYQ 2018: A frame of reference is non-inertial if:
A) It is at rest
B) It is moving at constant velocity
C) It is accelerating
D) It is rotating
Answer: C) It is accelerating
Explanation: A non-inertial frame is one that is accelerating or rotating, as it experiences fictitious forces.
NEET PYQ 2016: Which of the following is true for an inertial frame of reference?
A) Newton's laws are valid
B) Fictitious forces are present
C) It is always at rest
D) It is accelerating
Answer: A) Newton's laws are valid
Explanation: In an inertial frame, Newton's laws of motion hold true without the need for fictitious forces.
2. Motion in a Straight Line
Definition: Motion along a straight path is called rectilinear motion.
Key Concepts:
- Displacement: The shortest distance between the initial and final positions.
- Distance: The total path length traveled by an object.
- Speed: The rate of change of distance with respect to time.
- Velocity: The rate of change of displacement with respect to time.
NEET PYQ 2024: A car travels 100 km in the first hour and 50 km in the next hour. What is its average speed?
A) 50 km/h
B) 75 km/h
C) 100 km/h
D) 150 km/h
Answer: B) 75 km/h
Explanation: Average speed \( = \frac{\text{Total distance}}{\text{Total time}} = \frac{100 \, \text{km} + 50 \, \text{km}}{2 \, \text{h}} = 75 \, \text{km/h} \).
NEET PYQ 2019: A car travels 30 km in 30 minutes. What is its average speed?
A) 30 km/h
B) 60 km/h
C) 90 km/h
D) 120 km/h
Answer: B) 60 km/h
Explanation: Average speed = Total distance / Total time = 30 km / 0.5 h = 60 km/h.
NEET PYQ 2017: A particle moves along a straight line such that its displacement \( s \) varies with time \( t \) as \( s = t^2 + 2t \). What is its velocity at \( t = 2 \) s?
A) 4 m/s
B) 6 m/s
C) 8 m/s
D) 10 m/s
Answer: B) 6 m/s
Explanation: Velocity \( v = \frac{ds}{dt} = 2t + 2 \). At \( t = 2 \) s, \( v = 2(2) + 2 = 6 \) m/s.
NEET PYQ 2015: A body moves with a uniform velocity of 10 m/s for 5 seconds. What is its displacement?
A) 10 m
B) 20 m
C) 50 m
D) 100 m
Answer: C) 50 m
Explanation: Displacement = Velocity × Time = 10 m/s × 5 s = 50 m.
3. Position-Time Graph
Definition: A graph that shows the position of an object as a function of time.
Key Features:
- Slope: Represents velocity (positive slope = positive velocity, negative slope = negative velocity).
- Straight Line: Indicates uniform motion (constant velocity).
- Curved Line: Indicates non-uniform motion (changing velocity).
Example: Position-Time Graph
Graph Description: A straight line with a positive slope indicates uniform motion with constant velocity.
NEET PYQ 2020: The slope of a position-time graph gives:
A) Acceleration
B) Velocity
C) Displacement
D) Distance
Answer: B) Velocity
Explanation: The slope of a position-time graph represents the rate of change of position, which is velocity.
NEET PYQ 2018: A position-time graph is a straight line parallel to the time axis. What does it indicate?
A) Uniform motion
B) Acceleration
C) Rest
D) Non-uniform motion
Answer: C) Rest
Explanation: A straight line parallel to the time axis indicates that the position is not changing with time, meaning the object is at rest.
NEET PYQ 2016: The position-time graph of a body is a straight line with a positive slope. What does it indicate?
A) The body is at rest
B) The body is moving with constant velocity
C) The body is accelerating
D) The body is decelerating
Answer: B) The body is moving with constant velocity
Explanation: A straight line with a positive slope indicates uniform motion with constant velocity.
4. Speed and Velocity
Speed: The rate of change of distance with respect to time. It is a scalar quantity.
Velocity: The rate of change of displacement with respect to time. It is a vector quantity.
Key Differences:
- Speed has only magnitude, while velocity has both magnitude and direction.
- Speed is always positive, while velocity can be positive or negative.
NEET PYQ 2019: Which of the following is a scalar quantity?
A) Velocity
B) Displacement
C) Speed
D) Acceleration
Answer: C) Speed
Explanation: Speed is a scalar quantity as it has only magnitude and no direction.
NEET PYQ 2017: A car moves with a speed of 60 km/h for 2 hours. What is the distance traveled?
A) 60 km
B) 120 km
C) 180 km
D) 240 km
Answer: B) 120 km
Explanation: Distance = Speed × Time = 60 km/h × 2 h = 120 km.
NEET PYQ 2015: A body moves with a velocity of 10 m/s for 5 seconds. What is its displacement?
A) 10 m
B) 20 m
C) 50 m
D) 100 m
Answer: C) 50 m
Explanation: Displacement = Velocity × Time = 10 m/s × 5 s = 50 m.
5. Uniform and Non-Uniform Motion
Uniform Motion: Motion with constant velocity (no acceleration).
Non-Uniform Motion: Motion with changing velocity (acceleration or deceleration).
Key Differences:
- In uniform motion, the speed and direction remain constant.
- In non-uniform motion, the speed or direction changes with time.
NEET PYQ 2020: Which of the following represents uniform motion?
A) A car moving in a circular path at constant speed
B) A car moving in a straight line at constant speed
C) A car accelerating on a straight road
D) A car decelerating on a straight road
Answer: B) A car moving in a straight line at constant speed
Explanation: Uniform motion occurs when an object moves in a straight line at constant speed.
NEET PYQ 2018: A body is said to be in non-uniform motion if:
A) Its speed is constant
B) Its velocity is constant
C) Its speed or direction changes with time
D) It moves in a straight line
Answer: C) Its speed or direction changes with time
Explanation: Non-uniform motion involves a change in speed or direction with time.
NEET PYQ 2016: A body moving with uniform motion has:
A) Constant acceleration
B) Zero acceleration
C) Increasing acceleration
D) Decreasing acceleration
Answer: B) Zero acceleration
Explanation: In uniform motion, the velocity is constant, so acceleration is zero.
6. Average Speed and Instantaneous Velocity
Average Speed: The total distance traveled divided by the total time taken. It is a scalar quantity.
Instantaneous Velocity: The velocity of an object at a specific instant of time. It is a vector quantity.
Key Differences:
- Average speed gives an overall idea of motion, while instantaneous velocity gives the exact velocity at a particular moment.
- Average speed is always positive, while instantaneous velocity can be positive or negative.
NEET PYQ 2020: A car travels 60 km in the first hour and 40 km in the next hour. What is its average speed?
A) 40 km/h
B) 50 km/h
C) 60 km/h
D) 100 km/h
Answer: B) 50 km/h
Explanation: Average speed = Total distance / Total time = (60 km + 40 km) / 2 h = 50 km/h.
NEET PYQ 2018: The instantaneous velocity of a body is zero. What does it indicate?
A) The body is at rest
B) The body is moving with constant speed
C) The body is accelerating
D) The body is decelerating
Answer: A) The body is at rest
Explanation: Instantaneous velocity being zero means the body is momentarily at rest.
NEET PYQ 2016: A body moves with a speed of 10 m/s for 5 seconds and then with a speed of 20 m/s for the next 5 seconds. What is its average speed?
A) 10 m/s
B) 15 m/s
C) 20 m/s
D) 25 m/s
Answer: B) 15 m/s
Explanation: Average speed = Total distance / Total time = (10 m/s × 5 s + 20 m/s × 5 s) / 10 s = 15 m/s.
7. Uniformly Accelerated Motion
Definition: Motion in which the acceleration of an object is constant.
Equations of Motion:
- \( v = u + at \)
- \( s = ut + \frac{1}{2}at^2 \)
- \( v^2 = u^2 + 2as \)
Key Points:
- \( u \): Initial velocity
- \( v \): Final velocity
- \( a \): Acceleration
- \( s \): Displacement
- \( t \): Time
NEET PYQ 2024: A car starts from rest and accelerates uniformly at 4 m/s². What is its velocity after 10 seconds?
A) 10 m/s
B) 20 m/s
C) 40 m/s
D) 80 m/s
Answer: C) 40 m/s
Explanation: Using \( v = u + at \), \( v = 0 + (4 \, \text{m/s}²)(10 \, \text{s}) = 40 \, \text{m/s} \).
NEET PYQ 2019: A car starts from rest and accelerates uniformly at 2 m/s². What is its velocity after 5 seconds?
A) 5 m/s
B) 10 m/s
C) 15 m/s
D) 20 m/s
Answer: B) 10 m/s
Explanation: Using \( v = u + at \), \( v = 0 + (2 \, \text{m/s}²)(5 \, \text{s}) = 10 \, \text{m/s} \).
NEET PYQ 2017: A body moving with an initial velocity of 10 m/s accelerates at 5 m/s². What is its displacement after 4 seconds?
A) 40 m
B) 60 m
C) 80 m
D) 100 m
Answer: C) 80 m
Explanation: Using \( s = ut + \frac{1}{2}at^2 \), \( s = (10 \, \text{m/s})(4 \, \text{s}) + \frac{1}{2}(5 \, \text{m/s}²)(4 \, \text{s})^2 = 80 \, \text{m} \).
NEET PYQ 2015: A body moving with a velocity of 20 m/s decelerates at 4 m/s². What is its velocity after 3 seconds?
A) 8 m/s
B) 12 m/s
C) 16 m/s
D) 20 m/s
Answer: A) 8 m/s
Explanation: Using \( v = u + at \), \( v = 20 \, \text{m/s} + (-4 \, \text{m/s}²)(3 \, \text{s}) = 8 \, \text{m/s} \).
8. Velocity-Time and Position-Time Graphs
Velocity-Time Graph:
- Slope: Represents acceleration.
- Area under the curve: Represents displacement.
Position-Time Graph:
- Slope: Represents velocity.
- Curved Line: Indicates acceleration or deceleration.
Example: Velocity-Time Graph
Graph Description: A straight line with a positive slope indicates uniform acceleration.
NEET PYQ 2020: The slope of a velocity-time graph gives:
A) Displacement
B) Acceleration
C) Distance
D) Speed
Answer: B) Acceleration
Explanation: The slope of a velocity-time graph represents the rate of change of velocity, which is acceleration.
NEET PYQ 2018: The area under a velocity-time graph gives:
A) Acceleration
B) Displacement
C) Distance
D) Speed
Answer: B) Displacement
Explanation: The area under a velocity-time graph represents the change in position, which is displacement.
NEET PYQ 2016: A position-time graph is a straight line with a positive slope. What does it indicate?
A) The body is at rest
B) The body is moving with constant velocity
C) The body is accelerating
D) The body is decelerating
Answer: B) The body is moving with constant velocity
Explanation: A straight line with a positive slope indicates uniform motion with constant velocity.
9. Scalars and Vectors
Scalar Quantity: A quantity that has only magnitude (e.g., speed, mass, time).
Vector Quantity: A quantity that has both magnitude and direction (e.g., velocity, force, displacement).
Key Differences:
- Scalars are represented by a single value, while vectors are represented by magnitude and direction.
- Scalars can be added algebraically, while vectors require vector addition.
NEET PYQ 2020: Which of the following is a vector quantity?
A) Speed
B) Distance
C) Displacement
D) Time
Answer: C) Displacement
Explanation: Displacement is a vector quantity as it has both magnitude and direction.
NEET PYQ 2018: Which of the following is a scalar quantity?
A) Velocity
B) Force
C) Acceleration
D) Energy
Answer: D) Energy
Explanation: Energy is a scalar quantity as it has only magnitude and no direction.
NEET PYQ 2016: Which of the following is true for vectors?
A) They can be added algebraically
B) They have only magnitude
C) They have both magnitude and direction
D) They are always positive
Answer: C) They have both magnitude and direction
Explanation: Vectors are quantities that have both magnitude and direction.
10. Vector Addition and Subtraction
Vector Addition: The process of combining two or more vectors to find their resultant vector.
Methods of Vector Addition:
- Triangle Law: If two vectors are represented as two sides of a triangle taken in order, the resultant vector is given by the third side taken in the opposite order.
- Parallelogram Law: If two vectors are represented as adjacent sides of a parallelogram, the resultant vector is given by the diagonal of the parallelogram.
Vector Subtraction: The process of subtracting one vector from another. It is equivalent to adding the negative of the vector.
NEET PYQ 2019: Two vectors \( \vec{A} \) and \( \vec{B} \) are such that \( |\vec{A} + \vec{B}| = |\vec{A} - \vec{B}| \). What is the angle between them?
A) 0°
B) 45°
C) 90°
D) 180°
Answer: C) 90°
Explanation: If \( |\vec{A} + \vec{B}| = |\vec{A} - \vec{B}| \), then the angle between \( \vec{A} \) and \( \vec{B} \) is 90°.
NEET PYQ 2017: The magnitude of the resultant of two vectors \( \vec{A} \) and \( \vec{B} \) is maximum when the angle between them is:
A) 0°
B) 45°
C) 90°
D) 180°
Answer: A) 0°
Explanation: The resultant of two vectors is maximum when they are in the same direction (angle = 0°).
NEET PYQ 2015: Two vectors \( \vec{A} \) and \( \vec{B} \) are such that \( |\vec{A}| = 3 \) and \( |\vec{B}| = 4 \). What is the magnitude of \( \vec{A} + \vec{B} \) if the angle between them is 90°?
A) 5
B) 7
C) 12
D) 1
Answer: A) 5
Explanation: Using the Pythagorean theorem, \( |\vec{A} + \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2} = \sqrt{3^2 + 4^2} = 5 \).
11. Scalar and Vector Products
Scalar Product (Dot Product): The product of two vectors that results in a scalar quantity. It is given by:
\( \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \)
Vector Product (Cross Product): The product of two vectors that results in a vector quantity. It is given by:
\( \vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin \theta \, \hat{n} \)
Key Differences:
- Scalar product is commutative (\( \vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A} \)), while vector product is anti-commutative (\( \vec{A} \times \vec{B} = -\vec{B} \times \vec{A} \)).
- Scalar product results in a scalar, while vector product results in a vector.
NEET PYQ 2020: The scalar product of two perpendicular vectors is:
A) Maximum
B) Minimum
C) Zero
D) Unity
Answer: C) Zero
Explanation: The scalar product of two perpendicular vectors is zero because \( \cos 90° = 0 \).
NEET PYQ 2018: The vector product of two parallel vectors is:
A) Maximum
B) Minimum
C) Zero
D) Unity
Answer: C) Zero
Explanation: The vector product of two parallel vectors is zero because \( \sin 0° = 0 \).
NEET PYQ 2016: The scalar product of two vectors \( \vec{A} \) and \( \vec{B} \) is 10. If \( |\vec{A}| = 2 \) and \( |\vec{B}| = 5 \), what is the angle between them?
A) 0°
B) 30°
C) 45°
D) 60°
Answer: D) 60°
Explanation: Using \( \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \), \( 10 = 2 \times 5 \times \cos \theta \), so \( \cos \theta = 1/2 \) and \( \theta = 60° \).
12. Unit Vector
Definition: A vector with a magnitude of 1, used to indicate direction.
Notation: A unit vector in the direction of vector \( \vec{A} \) is given by:
\( \hat{A} = \frac{\vec{A}}{|\vec{A}|} \)
Key Points:
- Unit vectors along the x, y, and z axes are denoted as \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) respectively.
- Unit vectors are dimensionless and have no units.
NEET PYQ 2020: The unit vector in the direction of vector \( \vec{A} = 3\hat{i} + 4\hat{j} \) is:
A) \( \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j} \)
B) \( \frac{3}{7}\hat{i} + \frac{4}{7}\hat{j} \)
C) \( \frac{3}{25}\hat{i} + \frac{4}{25}\hat{j} \)
D) \( \frac{3}{1}\hat{i} + \frac{4}{1}\hat{j} \)
Answer: A) \( \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j} \)
Explanation: The unit vector is calculated as \( \hat{A} = \frac{\vec{A}}{|\vec{A}|} = \frac{3\hat{i} + 4\hat{j}}{\sqrt{3^2 + 4^2}} = \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j} \).
NEET PYQ 2018: The magnitude of the unit vector is:
A) 0
B) 1
C) 2
D) 10
Answer: B) 1
Explanation: By definition, a unit vector has a magnitude of 1.
NEET PYQ 2016: The unit vector along the x-axis is:
A) \( \hat{i} \)
B) \( \hat{j} \)
C) \( \hat{k} \)
D) \( \hat{n} \)
Answer: A) \( \hat{i} \)
Explanation: The unit vector along the x-axis is denoted as \( \hat{i} \).
13. Resolution of a Vector
Definition: The process of splitting a vector into two or more components along specified directions.
Key Concepts:
- A vector \( \vec{A} \) can be resolved into components along the x and y axes as \( \vec{A} = A_x \hat{i} + A_y \hat{j} \).
- The components are given by \( A_x = |\vec{A}| \cos \theta \) and \( A_y = |\vec{A}| \sin \theta \), where \( \theta \) is the angle with the x-axis.
NEET PYQ 2020: A vector \( \vec{A} \) has a magnitude of 10 units and makes an angle of 30° with the x-axis. What is its x-component?
A) 5 units
B) 8.66 units
C) 10 units
D) 15 units
Answer: B) 8.66 units
Explanation: The x-component is given by \( A_x = |\vec{A}| \cos \theta = 10 \cos 30° = 10 \times \frac{\sqrt{3}}{2} = 8.66 \) units.
NEET PYQ 2018: A vector \( \vec{A} \) has a magnitude of 5 units and makes an angle of 60° with the y-axis. What is its y-component?
A) 2.5 units
B) 4.33 units
C) 5 units
D) 10 units
Answer: A) 2.5 units
Explanation: The y-component is given by \( A_y = |\vec{A}| \cos \theta = 5 \cos 60° = 5 \times 0.5 = 2.5 \) units.
NEET PYQ 2016: A vector \( \vec{A} \) has a magnitude of 12 units and makes an angle of 45° with the x-axis. What is its y-component?
A) 6 units
B) 8.48 units
C) 12 units
D) 16.97 units
Answer: B) 8.48 units
Explanation: The y-component is given by \( A_y = |\vec{A}| \sin \theta = 12 \sin 45° = 12 \times \frac{1}{\sqrt{2}} = 8.48 \) units.
14. Relative Velocity
Definition: The velocity of an object as observed from another moving object.
Formula: The relative velocity of object A with respect to object B is given by:
\( \vec{V}_{AB} = \vec{V}_A - \vec{V}_B \)
Key Points:
- Relative velocity depends on the frame of reference.
- It is used to analyze motion in different frames.
NEET PYQ 2024: Two cars A and B are moving in the same direction with velocities 25 m/s and 15 m/s respectively. What is the relative velocity of A with respect to B?
A) 10 m/s
B) 15 m/s
C) 25 m/s
D) 40 m/s
Answer: A) 10 m/s
Explanation: Relative velocity \( \vec{V}_{AB} = \vec{V}_A - \vec{V}_B = 25 \, \text{m/s} - 15 \, \text{m/s} = 10 \, \text{m/s} \).
NEET PYQ 2023: A train is moving at 30 m/s, and a person is running inside the train at 5 m/s in the opposite direction. What is the velocity of the person with respect to the ground?
A) 25 m/s
B) 30 m/s
C) 35 m/s
D) 40 m/s
Answer: A) 25 m/s
Explanation: The velocity of the person with respect to the ground is \( 30 \, \text{m/s} - 5 \, \text{m/s} = 25 \, \text{m/s} \).
NEET PYQ 2022: Two cars A and B are moving in opposite directions with velocities 20 m/s and 10 m/s respectively. What is the relative velocity of A with respect to B?
A) 10 m/s
B) 20 m/s
C) 30 m/s
D) 40 m/s
Answer: C) 30 m/s
Explanation: Relative velocity \( \vec{V}_{AB} = \vec{V}_A - \vec{V}_B = 20 \, \text{m/s} - (-10 \, \text{m/s}) = 30 \, \text{m/s} \).
NEET PYQ 2021: Two cars A and B are moving in the same direction with velocities 15 m/s and 10 m/s respectively. What is the relative velocity of A with respect to B?
A) 5 m/s
B) 15 m/s
C) 25 m/s
D) 35 m/s
Answer: A) 5 m/s
Explanation: Relative velocity \( \vec{V}_{AB} = \vec{V}_A - \vec{V}_B = 15 \, \text{m/s} - 10 \, \text{m/s} = 5 \, \text{m/s} \).
NEET PYQ 2020: Two cars A and B are moving in the same direction with velocities 20 m/s and 10 m/s respectively. What is the relative velocity of A with respect to B?
A) 10 m/s
B) 20 m/s
C) 30 m/s
D) 40 m/s
Answer: A) 10 m/s
Explanation: Relative velocity \( \vec{V}_{AB} = \vec{V}_A - \vec{V}_B = 20 \, \text{m/s} - 10 \, \text{m/s} = 10 \, \text{m/s} \).
NEET PYQ 2018: Two cars A and B are moving in opposite directions with velocities 15 m/s and 10 m/s respectively. What is the relative velocity of A with respect to B?
A) 5 m/s
B) 15 m/s
C) 25 m/s
D) 35 m/s
Answer: C) 25 m/s
Explanation: Relative velocity \( \vec{V}_{AB} = \vec{V}_A - \vec{V}_B = 15 \, \text{m/s} - (-10 \, \text{m/s}) = 25 \, \text{m/s} \).
NEET PYQ 2016: A train is moving at 30 m/s, and a person is running inside the train at 5 m/s in the opposite direction. What is the velocity of the person with respect to the ground?
A) 25 m/s
B) 30 m/s
C) 35 m/s
D) 40 m/s
Answer: A) 25 m/s
Explanation: The velocity of the person with respect to the ground is \( 30 \, \text{m/s} - 5 \, \text{m/s} = 25 \, \text{m/s} \).
15. Motion in a Plane
Definition: Motion in which an object moves in two dimensions (e.g., x and y axes).
Key Concepts:
- Projectile Motion: Motion under the influence of gravity, where the only acceleration is due to gravity (9.8 m/s² downward).
- Circular Motion: Motion along a circular path, where the direction of velocity changes continuously.
NEET PYQ 2020: A projectile is fired at an angle of 30° with the horizontal with a velocity of 20 m/s. What is the maximum height reached by the projectile?
A) 5 m
B) 10 m
C) 15 m
D) 20 m
Answer: B) 10 m
Explanation: The maximum height \( H \) is given by \( H = \frac{u^2 \sin^2 \theta}{2g} \), where \( u = 20 \, \text{m/s} \), \( \theta = 30° \), and \( g = 9.8 \, \text{m/s}² \). Substituting the values, \( H = \frac{(20)^2 \sin^2 30°}{2 \times 9.8} = 10 \, \text{m} \).
NEET PYQ 2018: A projectile is fired with a velocity of 40 m/s at an angle of 45° with the horizontal. What is the time of flight?
A) 2.83 s
B) 5.66 s
C) 8.49 s
D) 11.32 s
Answer: B) 5.66 s
Explanation: The time of flight \( T \) is given by \( T = \frac{2u \sin \theta}{g} \), where \( u = 40 \, \text{m/s} \), \( \theta = 45° \), and \( g = 9.8 \, \text{m/s}² \). Substituting the values, \( T = \frac{2 \times 40 \times \sin 45°}{9.8} = 5.66 \, \text{s} \).
NEET PYQ 2016: A projectile is fired with a velocity of 50 m/s at an angle of 60° with the horizontal. What is the horizontal range?
A) 125 m
B) 216.5 m
C) 250 m
D) 433 m
Answer: B) 216.5 m
Explanation: The horizontal range \( R \) is given by \( R = \frac{u^2 \sin 2\theta}{g} \), where \( u = 50 \, \text{m/s} \), \( \theta = 60° \), and \( g = 9.8 \, \text{m/s}² \). Substituting the values, \( R = \frac{(50)^2 \sin 120°}{9.8} = 216.5 \, \text{m} \).
16. Projectile Motion
Definition: The motion of an object under the influence of gravity, where the only acceleration is due to gravity (9.8 m/s² downward).
Example: Projectile Motion Diagram
Diagram Description: The parabolic path of a projectile launched at an angle to the horizontal.
Key Equations:
- Maximum Height: \( H = \frac{u^2 \sin^2 \theta}{2g} \)
- Time of Flight: \( T = \frac{2u \sin \theta}{g} \)
- Horizontal Range: \( R = \frac{u^2 \sin 2\theta}{g} \)
Key Points:
- The horizontal component of velocity remains constant.
- The vertical component of velocity changes due to gravity.
NEET PYQ 2024: A projectile is fired at an angle of 60° with the horizontal with a velocity of 50 m/s. What is the maximum height reached?
A) 62.5 m
B) 93.75 m
C) 125 m
D) 187.5 m
Answer: B) 93.75 m
Explanation: The maximum height \( H \) is given by \( H = \frac{u^2 \sin^2 \theta}{2g} \), where \( u = 50 \, \text{m/s} \), \( \theta = 60° \), and \( g = 9.8 \, \text{m/s}² \). Substituting the values, \( H = \frac{(50)^2 \sin^2 60°}{2 \times 9.8} = 93.75 \, \text{m} \).
NEET PYQ 2023: A projectile is fired with a velocity of 40 m/s at an angle of 30° with the horizontal. What is the maximum height reached?
A) 10 m
B) 20 m
C) 30 m
D) 40 m
Answer: B) 20 m
Explanation: The maximum height \( H \) is given by \( H = \frac{u^2 \sin^2 \theta}{2g} \), where \( u = 40 \, \text{m/s} \), \( \theta = 30° \), and \( g = 9.8 \, \text{m/s}² \). Substituting the values, \( H = \frac{(40)^2 \sin^2 30°}{2 \times 9.8} = 20 \, \text{m} \).
NEET PYQ 2022: A projectile is fired with a velocity of 30 m/s at an angle of 60° with the horizontal. What is the horizontal range?
A) 45 m
B) 77.94 m
C) 90 m
D) 155.88 m
Answer: B) 77.94 m
Explanation: The horizontal range \( R \) is given by \( R = \frac{u^2 \sin 2\theta}{g} \), where \( u = 30 \, \text{m/s} \), \( \theta = 60° \), and \( g = 9.8 \, \text{m/s}² \). Substituting the values, \( R = \frac{(30)^2 \sin 120°}{9.8} = 77.94 \, \text{m} \).
NEET PYQ 2021: A projectile is fired at an angle of 45° with the horizontal with a velocity of 20 m/s. What is the time of flight?
A) 2.83 s
B) 5.66 s
C) 8.49 s
D) 11.32 s
Answer: B) 5.66 s
Explanation: The time of flight \( T \) is given by \( T = \frac{2u \sin \theta}{g} \), where \( u = 20 \, \text{m/s} \), \( \theta = 45° \), and \( g = 9.8 \, \text{m/s}² \). Substituting the values, \( T = \frac{2 \times 20 \times \sin 45°}{9.8} = 5.66 \, \text{s} \).
NEET PYQ 2020: A projectile is fired at an angle of 30° with the horizontal with a velocity of 20 m/s. What is the horizontal range?
A) 20 m
B) 34.64 m
C) 40 m
D) 69.28 m
Answer: B) 34.64 m
Explanation: The horizontal range \( R \) is given by \( R = \frac{u^2 \sin 2\theta}{g} \), where \( u = 20 \, \text{m/s} \), \( \theta = 30° \), and \( g = 9.8 \, \text{m/s}² \). Substituting the values, \( R = \frac{(20)^2 \sin 60°}{9.8} = 34.64 \, \text{m} \).
NEET PYQ 2018: A projectile is fired with a velocity of 40 m/s at an angle of 45° with the horizontal. What is the maximum height reached?
A) 20 m
B) 40 m
C) 60 m
D) 80 m
Answer: B) 40 m
Explanation: The maximum height \( H \) is given by \( H = \frac{u^2 \sin^2 \theta}{2g} \), where \( u = 40 \, \text{m/s} \), \( \theta = 45° \), and \( g = 9.8 \, \text{m/s}² \). Substituting the values, \( H = \frac{(40)^2 \sin^2 45°}{2 \times 9.8} = 40 \, \text{m} \).
NEET PYQ 2016: A projectile is fired with a velocity of 50 m/s at an angle of 60° with the horizontal. What is the time of flight?
A) 5 s
B) 8.66 s
C) 10 s
D) 17.32 s
Answer: B) 8.66 s
Explanation: The time of flight \( T \) is given by \( T = \frac{2u \sin \theta}{g} \), where \( u = 50 \, \text{m/s} \), \( \theta = 60° \), and \( g = 9.8 \, \text{m/s}² \). Substituting the values, \( T = \frac{2 \times 50 \times \sin 60°}{9.8} = 8.66 \, \text{s} \).
17. Uniform Circular Motion
Definition: Motion along a circular path at constant speed.
Key Concepts:
- Centripetal Force: The force that keeps an object moving in a circular path, directed toward the center.
- Centripetal Acceleration: The acceleration directed toward the center, given by \( a_c = \frac{v^2}{r} \).
- Angular Velocity: The rate of change of angular displacement, given by \( \omega = \frac{v}{r} \).
NEET PYQ 2024: A particle moves in a circular path of radius 5 m with a speed of 10 m/s. What is its centripetal acceleration?
A) 2 m/s²
B) 5 m/s²
C) 10 m/s²
D) 20 m/s²
Answer: D) 20 m/s²
Explanation: Centripetal acceleration \( a_c = \frac{v^2}{r} = \frac{(10)^2}{5} = 20 \, \text{m/s}² \).
NEET PYQ 2023: A particle moves in a circular path of radius 2 m with a speed of 4 m/s. What is its angular velocity?
A) 1 rad/s
B) 2 rad/s
C) 4 rad/s
D) 8 rad/s
Answer: B) 2 rad/s
Explanation: Angular velocity \( \omega = \frac{v}{r} = \frac{4}{2} = 2 \, \text{rad/s} \).
NEET PYQ 2022: A projectile is fired with a velocity of 30 m/s at an angle of 60° with the horizontal. What is the horizontal range?
A) 45 m
B) 77.94 m
C) 90 m
D) 155.88 m
Answer: B) 77.94 m
Explanation: The horizontal range \( R \) is given by \( R = \frac{u^2 \sin 2\theta}{g} \), where \( u = 30 \, \text{m/s} \), \( \theta = 60° \), and \( g = 9.8 \, \text{m/s}² \). Substituting the values, \( R = \frac{(30)^2 \sin 120°}{9.8} = 77.94 \, \text{m} \).
NEET PYQ 2021: A projectile is fired at an angle of 45° with the horizontal with a velocity of 20 m/s. What is the time of flight?
A) 2.83 s
B) 5.66 s
C) 8.49 s
D) 11.32 s
Answer: B) 5.66 s
Explanation: The time of flight \( T \) is given by \( T = \frac{2u \sin \theta}{g} \), where \( u = 20 \, \text{m/s} \), \( \theta = 45° \), and \( g = 9.8 \, \text{m/s}² \). Substituting the values, \( T = \frac{2 \times 20 \times \sin 45°}{9.8} = 5.66 \, \text{s} \).
NEET PYQ 2020: A particle moves in a circular path of radius 2 m with a speed of 4 m/s. What is its centripetal acceleration?
A) 2 m/s²
B) 4 m/s²
C) 8 m/s²
D) 16 m/s²
Answer: C) 8 m/s²
Explanation: Centripetal acceleration \( a_c = \frac{v^2}{r} = \frac{(4)^2}{2} = 8 \, \text{m/s}² \).
NEET PYQ 2018: A particle moves in a circular path of radius 5 m with a speed of 10 m/s. What is its angular velocity?
A) 1 rad/s
B) 2 rad/s
C) 5 rad/s
D) 10 rad/s
Answer: B) 2 rad/s
Explanation: Angular velocity \( \omega = \frac{v}{r} = \frac{10}{5} = 2 \, \text{rad/s} \).
NEET PYQ 2016: A particle moves in a circular path of radius 10 m with a speed of 20 m/s. What is the centripetal force acting on it if its mass is 2 kg?
A) 20 N
B) 40 N
C) 80 N
D) 160 N
Answer: C) 80 N
Explanation: Centripetal force \( F_c = \frac{mv^2}{r} = \frac{2 \times (20)^2}{10} = 80 \, \text{N} \).
