Key Notes:

Understanding Graphical Representation:

• The equations of motion describe the relationship between displacement, velocity, acceleration, and time.
• These relationships can be represented graphically using velocity-time (v-t) graphs and displacement-time (s-t) graphs.

Types of Graphs:

• Velocity-Time (v-t) Graph: Shows how velocity changes with time.
  • The slope of a velocity-time graph gives acceleration.
  • The area under the v-t graph gives displacement.
• Displacement-Time (s-t) Graph: Shows how displacement changes with time.
  • The slope of a displacement-time graph gives velocity.
  • If the graph is a straight line, it indicates uniform motion.

Equations of Motion:

• The three fundamental equations of motion for uniformly accelerated motion are:
  1. v = u + at
     • v = final velocity
     • u = initial velocity
     • a = acceleration
     • t = time
  2. s = ut + ½at²
     • s = displacement
     • u = initial velocity
     • a = acceleration
     • t = time
  3. v² = u² + 2as
     • v = final velocity
     • u = initial velocity
     • a = acceleration
     • s = displacement

Graphical Method of Deriving the Equations:

• Velocity-Time Graph:
  • For uniform acceleration, the graph is a straight line with a positive or negative slope (depending on the direction of acceleration).
  • The equation v = u + at can be derived by observing the slope of the velocity-time graph.
• Displacement-Time Graph:
  • The area under a v-t graph represents displacement, and this can be used to derive s = ut + ½at².
• Slope and Area Interpretation:
  • The slope of the displacement-time graph gives the instantaneous velocity.
  • The area under the velocity-time graph gives the total displacement.

Derivation of Equations Using Graphs:

• First Equation: From a v-t graph, the slope represents acceleration, so a = (v – u)/t. Rearranging gives v = u + at.
• Second Equation: From the area under the v-t graph, the displacement is given by the area of the trapezoid, which simplifies to s = ut + ½at².
• Third Equation: By considering the kinematic relations and the area under the velocity-time graph, we can derive v² = u² + 2as.

Applications:

• These equations are used to solve problems related to motion in a straight line, like calculating final velocity, displacement, and time when acceleration is constant.

Example Problems:

• Calculate the displacement of a body given its velocity-time graph.
• Use the graphical method to find the acceleration from a velocity-time graph.

Let’s practice!