Motion in Plane

Motion in a Plane

Join Our Study Groups

Scalar and Vector Quantities

Properties & Key Points:

  • Scalar quantities have only magnitude (e.g., distance, speed, temperature).
  • Vector quantities have both magnitude and direction (e.g., displacement, velocity, acceleration).
Join Our Study Groups

Position Vector

\[\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}\]

Properties & Key Points:

  • The position vector of an object is a vector from the origin to the point where the object is located.
  • \( \vec{r} \): Position vector.
  • \( x, y, z \): Coordinates of the object in the x, y, and z directions.
  • \( \hat{i}, \hat{j}, \hat{k} \): Unit vectors along the x, y, and z axes.
Join Our Study Groups

Displacement

\[\vec{d} = \vec{r}_2 - \vec{r}_1\]

Properties & Key Points:

  • Displacement is the change in position of an object and is a vector quantity.
  • \( \vec{d} \): Displacement vector.
  • \( \vec{r}_1, \vec{r}_2 \): Initial and final position vectors.
Join Our Study Groups

Velocity

\[\vec{v} = \frac{\vec{d}}{t}\]

Properties & Key Points:

  • Velocity is the rate of change of displacement with respect to time. It is a vector quantity.
  • \( \vec{v} \): Velocity vector.
  • \( \vec{d} \): Displacement.
  • \( t \): Time.
Join Our Study Groups

Speed

\[v = \frac{d}{t}\]

Properties & Key Points:

  • Speed is the rate of change of distance with respect to time and is a scalar quantity.
  • \( v \): Speed.
  • \( d \): Distance traveled.
  • \( t \): Time.
Join Our Study Groups

Acceleration

\[\vec{a} = \frac{\vec{v} - \vec{u}}{t}\]

Properties & Key Points:

  • Acceleration is the rate of change of velocity with respect to time. It is a vector quantity.
  • \( \vec{a} \): Acceleration vector.
  • \( \vec{v} \): Final velocity.
  • \( \vec{u} \): Initial velocity.
  • \( t \): Time.
Join Our Study Groups

Motion in Two Dimensions

Properties & Key Points:

  • For motion in a plane, the motion can be described in terms of the x and y components of position, velocity, and acceleration.
  • The horizontal and vertical motions are independent of each other.
Join Our Study Groups

Kinematic Equations for Uniformly Accelerated Motion

Properties & Key Points:

  • In the x-direction (horizontal):
  • In the y-direction (vertical):
  • \( v_x, v_y \): Final velocity in the x and y directions.
  • \( u_x, u_y \): Initial velocity in the x and y directions.
  • \( a_x, a_y \): Acceleration in the x and y directions.
  • \( x, y \): Displacement in the x and y directions.
  • \( t \): Time.
Join Our Study Groups

Projectile Motion

Properties & Key Points:

  • Projectile motion is the motion of an object thrown into the air under the influence of gravity. The motion can be split into horizontal and vertical components.
  • The horizontal velocity remains constant, and the vertical velocity is affected by gravity.
Join Our Study Groups

Horizontal Motion in Projectile Motion

Properties & Key Points:

  • Horizontal velocity is constant:
  • Horizontal displacement:
Join Our Study Groups

Vertical Motion in Projectile Motion

Properties & Key Points:

  • Vertical velocity:
  • Vertical displacement:
  • Maximum height:
  • Time of flight:
  • Range of the projectile:
  • \( u_x, u_y \): Initial horizontal and vertical velocity.
  • \( v_x, v_y \): Final horizontal and vertical velocity.
  • \( g \): Acceleration due to gravity.
  • \( x, y \): Horizontal and vertical displacement.
  • \( t \): Time.
Join Our Study Groups

Relative Motion

\[\vec{v}_{AB} = \vec{v}_A - \vec{v}_B\]

Properties & Key Points:

  • Relative velocity is the velocity of an object with respect to another object.
  • \( \vec{v}_{AB} \): Velocity of object A relative to object B.
  • \( \vec{v}_A, \vec{v}_B \): Velocity of objects A and B, respectively.
Join Our Study Groups

Circular Motion

\[\theta = \frac{s}{r}\]
\[\omega = \frac{\theta}{t}\]
\[\alpha = \frac{\omega - \omega_0}{t}\]
\[v_t = r \omega\]
\[a_c = \frac{v_t^2}{r} = r \omega^2\]

Properties & Key Points:

  • Angular displacement is the angle through which an object moves on a circular path.
  • \( \theta \): Angular displacement.
  • \( s \): Arc length (distance traveled along the circle).
  • \( r \): Radius of the circle.
  • Angular velocity is the rate of change of angular displacement.
  • \( \omega \): Angular velocity.
  • \( \theta \): Angular displacement.
  • \( t \): Time.
  • Angular acceleration is the rate of change of angular velocity.
  • \( \alpha \): Angular acceleration.
  • \( \omega_0 \): Initial angular velocity.
  • Tangential velocity is the linear velocity of an object moving along a circular path.
  • \( v_t \): Tangential velocity.
  • \( r \): Radius of the circle.
  • \( \omega \): Angular velocity.
  • Centripetal acceleration is the acceleration directed toward the center of the circular path.
  • \( a_c \): Centripetal acceleration.
  • \( v_t \): Tangential velocity.
  • \( r \): Radius of the circle.
Join Our Study Groups