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Motion in Two Dimensions in Physics

Once motion leaves the straight line, position, velocity and acceleration stop being single numbers and each becomes a pair of components that can be handled independently. That one idea does most of the work on this page. It turns a curved flight into two straight-line problems, it explains why the horizontal velocity of a projectile never changes while the vertical one does, and it is the reason a body moving at constant speed round a circle is still accelerating. The blocks follow the topic in order, from the vectors themselves through projectile flight and circular motion to the relative motion of two observers.

Position vector of a point in a plane

Given

  • (m)
Displacement vector between two positions

Given

  • (m)
  • (m)
Average velocity vector

Given

  • (m)
  • (s)
Instantaneous velocity vector

Given

Average acceleration vector

Given

  • (m/s)
  • (s)
Instantaneous acceleration vector

Given

Velocity vector written from its components

Given

  • (m/s)
Acceleration vector written from its components

Given

  • (m/s²)
Speed from the velocity components

Given

  • (m/s)
Direction of the velocity from its components

Given

  • (m/s)
Velocity after a time under constant acceleration

Given

  • (m/s)
  • (m/s²)
  • (s)
Displacement after a time under constant acceleration

Given

  • (m/s)
  • (m/s²)
  • (s)
Velocity from the displacement, without needing the time

Given

  • (m/s)
  • (m/s²)
  • (m)
Displacement from the mean of the two velocities

Given

  • (m/s)
  • (m/s)
  • (s)
Horizontal displacement of a projectile at a time

Given

  • (m/s)
  • (rad)
  • (s)
Vertical displacement of a projectile at a time

Given

  • (m/s)
  • (rad)
  • (s)

Constant used

Velocity components of a projectile at a time

Given

  • (m/s)
  • (rad)
  • (s)

Constant used

Speed of a projectile at a time

Given

  • (m/s)
  • (m/s)
Direction of a projectile velocity at a time

Given

  • (m/s)
Time for a projectile to reach its greatest height

Given

  • (m/s)
  • (rad)

Constant used

Greatest height reached by a projectile

Given

  • (m/s)
  • (rad)

Constant used

Time of flight back to the launch height

Given

  • (m/s)
  • (rad)

Constant used

Horizontal range on level ground

Given

  • (m/s)
  • (rad)

Constant used

Greatest range possible for a given launch speed

Given

  • (m/s)

Constant used

Time to fall from a height when launched horizontally

Given

  • (m)

Constant used

Horizontal distance covered by a horizontal launch

Given

  • (m/s)
  • (m)

Constant used

Speed at impact from a horizontal launch

Given

  • (m/s)
  • (m)

Constant used

Period of a body in uniform circular motion

Given

  • (m)
  • (m/s)
Speed from the period of a circular motion

Given

  • (m)
  • (s)
Angular speed from the period

Given

  • (s)
Speed from the angular speed

Given

  • (rad/s)
  • (m)
Frequency from the period

Given

  • (s)
Angular speed from the frequency

Given

  • (Hz)
Acceleration in uniform circular motion

Given

  • (m/s)
  • (m)
  • (rad/s)
Force needed to keep a body in uniform circular motion

Given

  • (kg)
  • (m/s)
  • (m)
Tangential acceleration when the speed is changing

Given

  • (s)
Total acceleration when the speed is changing

Given

  • (m/s)
  • (m)
  • (m/s²)
Direction of the total acceleration

Given

  • (m/s²)
  • (m/s²)
Velocity of one body relative to another

Given

  • (m/s)
  • (m/s)
Acceleration of one body relative to another

Given

  • (m/s²)
  • (m/s²)
Combining velocities across a moving frame

Given

  • (m/s)
  • (m/s)
Time to cross a river aimed straight across

Given

  • (m)
  • (m/s)
Downstream drift while crossing a river

Given

  • (m/s)
  • (s)
Resultant speed while crossing a river

Given

  • (m/s)
  • (m/s)