Motion in a Plane: Quick Revision Sheet

 Key Concepts

• Scalar Quantity: A quantity with magnitude only (e.g., mass, speed, time).
• Vector Quantity: A quantity with magnitude and direction (e.g., displacement, velocity, acceleration).
• Displacement: The change in position of an object, measured as a vector.
• Velocity: The rate of change of displacement, measured as a vector.
• Acceleration: The rate of change of velocity, measured as a vector.
• Projectile Motion: The motion of an object under the influence of gravity only.
• Uniform Circular Motion: The motion of an object in a circle with constant speed.
• Centripetal Force: The force that keeps an object moving in a circle.
• Centrifugal Force: A fictitious force that appears to act outward on an object moving in a circle.

Equations

• Velocity: v = ds/dt
• Acceleration: a = dv/dt
• Projectile Motion:
  • Horizontal component: x = u_x t
  • Vertical component: y = u_y t - (1/2)gt^2
• Uniform Circular Motion:
  • Speed: v = 2πr/T
  • Centripetal acceleration: a_c = v^2/r = 4π^2r/T^2
  • Centripetal force: F_c = mv^2/r = 4π^2mr/T^2

Important Points

• The horizontal and vertical components of projectile motion are independent of each other.
• The time of flight of a projectile depends only on its initial vertical velocity.
• The maximum height reached by a projectile depends on its initial vertical velocity and the acceleration due to gravity.
• In uniform circular motion, the direction of the velocity vector is always tangent to the circle.
• The centripetal force is always directed towards the center of the circle.

Problems to Practice

1. A ball is thrown horizontally from the top of a building with a speed of 20 m/s. If the building is 50 m tall, how far from the base of the building will the ball land?
2. A car is moving around a circular track of radius 100 m with a speed of 30 m/s. What is the centripetal acceleration of the car?
3. A stone is thrown vertically upward with a speed of 30 m/s. How long will it take to reach its maximum height?