MOTION IN A PLANE

1. Scalar and Vector Quantities:

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   Scalars are quantities that have magnitude only, such as mass, temperature, time, and distance. They are completely described by a numerical value and a unit. For example, the mass of an object is a scalar quantity.

Vectors, on the other hand, have both magnitude and direction. They are represented by arrows, where the length of the arrow represents the magnitude and the direction of the arrow represents the direction of the vector. Examples of vector quantities include displacement, velocity, force, and acceleration.

2. Position and Displacement Vectors:
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2. The position vector, denoted as r, represents the position of a point in space relative to a reference point or origin. It has both magnitude and direction. For example, if a point P is located 3 units to the right and 2 units above the origin, the position vector of P is (3, 2).

The displacement vector, denoted as Δr, represents the change in position of an object. It is the difference between the final position vector and the initial position vector. For example, if an object moves from point A to point B, the displacement vector Δr is given by Δr = r_B - r_A.

3. General Vectors and their Notations: In general, a vector is represented by a symbol with an arrow on top, such as v⃗, a⃗, or F⃗, to distinguish it from scalar quantities. The arrow indicates the direction of the vector.

4. Equality of Vectors: Two vectors are considered equal if they have the same magnitude and the same direction. It means that for two vectors v⃗ and u⃗ to be equal, |v⃗| = |u⃗|, and they must point in the same direction.

5. Addition and Subtraction of Vectors: Vector addition is performed by placing the tail of the second vector at the head of the first vector and drawing a new vector from the tail of the first vector to the head of the second vector. The resultant vector is the vector that goes from the tail of the first vector to the head of the second vector. For example, if vector A⃗ = (3, 2) and vector B⃗ = (-1, 4), their sum A⃗ + B⃗ is (3, 2) + (-1, 4) = (2, 6).

Vector subtraction is similar to addition but involves reversing the direction of the vector being subtracted. For example, if vector A⃗ = (3, 2) and vector B⃗ = (-1, 4), their difference A⃗ - B⃗ is (3, 2) - (-1, 4) = (4, -2).

6. Relative Velocity:

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   Relative velocity is the velocity of an object or observer in the frame of reference of another object or observer. It represents the velocity of one object relative to another. The relative velocity between two objects can be found by subtracting their individual velocities. For example, if a person is walking at 3 m/s in the forward direction on a moving train, and the train is moving at 10 m/s in the forward direction, then the relative velocity of the person with respect to the ground is 3 m/s + 10 m/s = 13 m/s in the forward direction.

8. Unit Vector: A unit vector is a vector that has a magnitude of 1. It is used to represent direction. A unit vector is denoted by placing a caret (^) on top of the vector symbol. For example, ȳ⃗ is a unit vector in the y-direction.

9. Resolution of a Vector in a Plane: Resolution of a vector refers to breaking it down into its components along two or more perpendicular axes. The rectangular coordinate system, with x and y axes, is commonly used for this purpose. By using trigonometry and the Pythagorean theorem, a vector can be resolved into its components.

10. Rectangular Components: Rectangular components are the magnitudes of a vector along the x and y axes. They are usually represented as (x, y) or (i, j), where i⃗ is the unit vector in the x-direction and j⃗ is the unit vector in the y-direction. For example, if a vector v⃗ has a magnitude of 5 units and is inclined at an angle of 30 degrees with the x-axis, its rectangular components would be v_x = 5 cos(30°) and v_y = 5 sin(30°).

11. Scalar and Vector Product of Two Vectors: Scalar product, also known as dot product, is a binary operation that takes two vectors and returns a scalar quantity. It is calculated by multiplying the magnitudes of the vectors and the cosine of the angle between them. The scalar product of vectors A⃗ and B⃗ is denoted as A⃗ · B⃗ or A⃗ ⋅ B⃗.

Vector product, also known as cross product, is a binary operation that takes two vectors and returns a vector perpendicular to both vectors. It is calculated by multiplying the magnitudes of the vectors, the sine of the angle between them, and the unit vector perpendicular to the plane containing the two vectors. The vector product of vectors A⃗ and B⃗ is denoted as A⃗ × B⃗.

11. Projectile Motion:

    Projectile motion refers to the motion of an object projected into the air, under the influence of gravity, with no other forces acting on it. The object follows a curved path called a trajectory. The motion can be analyzed by breaking it down into horizontal and vertical components. An example of projectile motion is a ball thrown into the air.

12. Uniform Circular Motion:

    Uniform circular motion occurs when an object moves in a circular path at a constant speed. The object experiences a centripetal force directed toward the center of the circle. Examples of uniform circular motion include a satellite orbiting the Earth or a car moving around a circular track at a constant speed.

1. MCQ WORK SHEET 1 (JEE/NEET)
2. MCQ WORK SHEET 2 (JEE/NEET)
3. MCQ WORK SHEET 3 (JEE/NEET)
4. MCQ WORK SHEET 4 (JEE/NEET)
5. MCQ WORK SHEET 5 (JEE/NEET)
6. MCQ WORK SHEET 1 (GUJ)
7. MCQ WORK SHEET 2 (GUJ)