Kinematics

This section focusses on kinematics. Kinematics is the study of motion without considering the forces that cause it. In the context of Mechanics, it primarily involves understanding the relationships between displacement, velocity, and acceleration for objects moving in a straight line under constant acceleration.

Displacement, Velocity, and Acceleration

• Displacement (s):
  Displacement is a vector quantity, representing the straight-line distance from an object's initial position to its final position, with a direction.
• Velocity (v):
  Velocity is a vector quantity that measures the rate of change of displacement. It has both magnitude (speed) and direction.
  • Average velocity: vˉ=Δs/Δt where Δs is the change in displacement and Δt  is the change in time.
  • Instantaneous velocity: The velocity at any given moment in time.
• Acceleration (a):
  Acceleration is a vector quantity that measures the rate of change of velocity. It can be uniform (constant) or variable.
  For constant acceleration:
  a=Δv/Δt

SUVAT Equations

The SUVAT equations are a set of five key equations used to describe motion under constant acceleration. They link displacement, initial velocity, final velocity, acceleration, and time.

Let:

• s = displacement
• u = initial velocity
• v = final velocity
• a = acceleration
• t = time

The five SUVAT equations are:

v=u+at

• Final velocity = initial velocity + (acceleration × time)
• Used to find the final velocity when initial velocity, acceleration, and time are known.

s=ut+½at²

• Displacement = (initial velocity × time) + ½ (acceleration × time²)
• Used to find displacement when initial velocity, acceleration, and time are known.

v²=u²+2as

• Final velocity² = initial velocity² + 2 × (acceleration × displacement)
• Used to find the final velocity when initial velocity, acceleration, and displacement are known.

s=u+v/2×t

• Displacement = average velocity × time
• Used to find displacement when the initial velocity, final velocity, and time are known.

s=vt−½at²

• Displacement = (final velocity × time) − ½​ (acceleration × time²)
• Another form of the equation to calculate displacement when final velocity, acceleration, and time are known.

Understanding the SUVAT Equations

Each equation is useful for different types of problems, depending on which quantities are given. The most common types of questions involve:

• Finding one unknown (displacement, velocity, or time) when the other quantities are known.
• Dealing with objects moving in a straight line with constant acceleration, such as free-falling objects (ignoring air resistance) or objects in uniform motion.

Application to Real-World Problems

To use the SUVAT equations correctly, always:

1. Identify the given quantities: Determine the values for u, v, a, s, and t from the problem statement.
2. Select the appropriate SUVAT equation: Based on the known values, choose the equation that involves the unknown quantity.
3. Substitute the known values into the equation and solve for the unknown.
4. Check the units: Ensure the units for acceleration, velocity, displacement, and time are consistent (e.g., m/s² for acceleration, m/s for velocity, and m for displacement).

Common Problem Types

Example: A car accelerates from rest

A car starts from rest and accelerates at a constant rate of 2 m/s² for 5 seconds. Find its final velocity and the distance it travels.

Given:
u=0 m/s, a=2 m/s², t=5 s

Find:
v (final velocity), s (displacement)

Use equation 1: v=u+at

Use equation 2: s=ut+½at²

Final velocity = 10 m/s, Displacement = 25 m

Graphical Representation

• Displacement-Time Graph:
  For uniform motion, the graph is a straight line with a constant gradient, representing constant velocity. For accelerated motion, the graph is a curve, with the gradient increasing over time.
• Velocity-Time Graph:
  For uniform motion, the graph is a horizontal line. For motion with constant acceleration, the graph is a straight line with a constant slope (equal to acceleration). The area under a velocity-time graph represents the displacement.
• Acceleration-Time Graph:
  For motion with constant acceleration, the graph is a horizontal line. If the acceleration is zero, the object moves with uniform velocity.

Key Points to Remember

• Ensure that the units for time, velocity, displacement, and acceleration are consistent (usually metres, seconds, and metres per second).
• When solving problems, always check which quantities are given and which one you need to find, then select the correct SUVAT equation.
• Practice different types of problems, including those involving free-falling objects, projectiles, and objects with uniform acceleration.

By mastering the relationships between displacement, velocity, and acceleration, and becoming comfortable with the SUVAT equations, you'll be well-equipped to tackle a wide range of problems in kinematics.