DP IB Physics: SL

  1. Space, time and motion

A.1 Kinematics

DP IB Physics: SL

A. Space, time and motion

A.1 Kinematics

Understandings
Standard level and higher level: 9 hours
Students should understand:
a) That the motion of bodies through space and time can be described and analyzed in terms of position, velocity, and acceleration
b) Velocity is the rate of change of position, and acceleration is the rate of change of velocity
c) The change in position is the displacement
d) The difference between distance and displacement
e) The difference between instantaneous and average values of velocity, speed and acceleration, and how to determine them
f) The equations of motion for solving problems with uniformly accelerated motion as given by

 [math]s = \frac{u + v}{2} \cdot t \\
v = u + at \\
s = ut + \frac{1}{2}at^2[/math]
g) Motion with uniform and non-uniform acceleration
h) The behavior of projectiles in the absence of fluid resistance, and the application of the equations of motion resolved into vertical and horizontal components
i) The qualitative effect of fluid resistance on projectiles, including time of flight, trajectory, velocity, acceleration, range and terminal speed.

  • a) Motion of Bodies Through Space and Time

  • Objects move in space, and their motion can be tracked over time. The motion is analyzed using kinematics, which studies motion without considering the forces causing it.
  • ⇒ Concepts of Motion:
    1. Position (x): The location of an object in space relative to a reference point.
    2. Velocity (v): The rate of change of position. It tells how fast and in what direction an object moves.
    3. Acceleration (a): The rate of change of velocity. It shows how an object’s speed or direction changes.
  • Objects can experience different types of motion:
  • – Uniform motion: Constant velocity (no acceleration).
  • – Non-uniform motion: Changing velocity (acceleration present).
  • – Linear motion: Motion along a straight line.
  • Projectile motion: Motion under the influence of gravity.

  • b) Velocity and Acceleration

  • ⇒  Velocity (Rate of Change of Position):
  • Velocity (v) is defined as the rate of change of position with respect to time. It is a vector quantity, meaning it has both magnitude and direction.
  • Figure 1 Velocity and acceleration of a car
  • [math]v = \frac{\Delta x}{\Delta t}[/math]
  • Where:
  • – v = velocity
  • – [math]\Delta x[/math] = change in position (displacement)
  • – [math]\Delta t[/math] = change in time
  • ⇒ Instantaneous Velocity:
  • The velocity at a specific instant in time. It is found using calculus:
  • [math]v = \frac{dx}{dt}[/math]
  • ⇒ Average Velocity:
  • The total displacement divided by total time.
  • [math]v_{\text{avg}} = \frac{x_2 – x_1}{t_2 – t_1}[/math]
  • If velocity changes over time, there is acceleration.
  • ⇒  Acceleration (Rate of Change of Velocity):
  • Acceleration (a) is the rate of change of velocity with respect to time. It describes how quickly an object speeds up or slows down.
  • [math]a = \frac{\Delta v}{\Delta t}[/math]
  • Where:
  • – a = acceleration
  • – [math]\Delta v[/math]= change in velocity
  • – [math]\Delta t[/math] = change in time
  • ⇒ Instantaneous Acceleration:
  • The acceleration at a specific moment:
  • [math]a = \frac{dv}{dt}[/math]
  • ⇒ Uniform Acceleration:
  • Acceleration remains constant over time (e.g., free fall under gravity, g = 9.8 m/s2).
  • ⇒ Non-Uniform Acceleration:
  • Acceleration changes over time (e.g., a car increasing speed at varying rates).

  • c) Displacement (Change in Position):

  • Displacement ([math]\Delta x[/math]) is the change in position of an object. It is a vector quantity, meaning it has both magnitude and direction.
  • [math]\Delta x = x_2 – x_1[/math]
  • Where:
  • [math]x_2[/math]​ = final position
  • [math]x_1[/math]​ = initial position
  • Figure 2 Displacement between two points A and B
  • ⇒ Difference Between Displacement and Distance:
  • Displacement considers only the shortest straight-line path from start to end.
  • Distance is the total length of the path traveled, regardless of direction.
  • For example:
  • A runner moves 10 m forward and then 10 m back.
  • Distance = 20 m
  • Displacement = 0 m (since the start and end points are the same).
  • ⇒ Equations of Motion (Kinematic Equations)
  • For objects moving with constant acceleration:
  • 1. Velocity-time relation:
  • [math]v = u + at[/math]
  • 2. Displacement-time relation:
  • [math]s = ut + \frac{1}{2} at^2[/math]
  • 3. Velocity-displacement relation:
  • [math]v^2 = u^2 + 2as[/math]
  • Where:
  • – u = initial velocity
  • – v = final velocity
  • – a = acceleration
  • – s = displacement
  • – t = time
  • Motion is described using different quantities, such as distance, displacement, speed, velocity, and acceleration. These are fundamental concepts in kinematics, which deals with motion without considering forces.

  • d) Distance vs. Displacement

Distance Displacement
The total path covered by an object. The shortest straight-line distance from the starting point to the endpoint.
Scalar quantity (has only magnitude). Vector quantity (has both magnitude and direction).
Always positive or zero. Can be positive, negative, or zero.
Example: A car moves 4 m forward and 3 m backward. Distance = 4 + 3 = 7 m. Example: A car moves 4 m forward and 3 m backward. Displacement = 4 – 3 = 1 m forward.
  • Distance is always greater than or equal to displacement.
  • If an object returns to its starting point, distance > 0 but displacement = 0.

  • e) Instantaneous vs. Average Values

  • ⇒  Instantaneous Values
  • Instantaneous Speed:
  • The speed of an object at a specific moment in time.
  • Measured using speedometers or calculated using calculus ([math]v = \frac{dx}{dt}[/math]).
  • Instantaneous Velocity:
  • The velocity at a specific instant, considering direction.
  • Figure 3 Instantaneous velocity graph between position and time
  • Instantaneous Acceleration:
  • The acceleration at a particular instant ([math]a = \frac{dv}{dt}[/math]​).
  • Figure 4 Instantons velocity
  • ⇒ Average Values
  • [math]\begin{gather}
    \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \\
    v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} \\
    \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} \\
    v_{\text{avg}} = \frac{\text{Total Displacement}}{\text{Total Time}} \\
    \text{Average Acceleration} = \frac{\text{Change in Velocity}}{\text{Time Taken}} \\
    a = \frac{v – u}{t}
    \end{gather} [/math]
  • ⇒ Example:
  • A car travels 100 m in 10 s.
  • [math]\text{Average Speed} = \frac{100}{10} = 10 \, \text{m/s}[/math]
  • If the car starts at 0 m and stops at 50 m,
  • [math]\text{Average velocity} = 50/10 = 5 m/s[/math]
  • ⇒  Differences:
Instantaneous Average
Measured at a single moment. Measured over a time interval.
Requires calculus for exact values. Simple formula-based calculations.

  • f) Equations of Motion

  • For uniformly accelerated motion (constant acceleration), we use three kinematic equations:
  • 1. Equation for Displacement Using Average Velocity:
  • [math]s = \frac{(u + v)}{2} \cdot t[/math]
  • – u = Initial velocity
  • – v = Final velocity
  • – s = Displacement
  • – t = Time taken
  • ⇒ Example:
  • A car accelerates from 5 m/s to 15 m/s in 4 s. Find the displacement.
  • [math]s = \frac{5 + 15}{2} \times 4 \\
    s = \frac{20}{2} \times 4 = 40 \, \text{m}[/math]
  • 2. Velocity-Time Relation:
  • [math]v = u + at[/math]
  • a = Acceleration
  • ⇒ Example:
    A car starts at rest (u=0) and accelerates at [math]2m/s^2[/math] for 5 Find final velocity.
  • [math]v = u + at \\
    v = 0 + (2 \times 5) \\
    v = 10 \, \text{m/s}[/math]
  • 3. Displacement-Time Relation:
  • [math]s = ut + \frac{1}{2} a t^2[/math]
  • ⇒ Example:
  • A car starts from rest (u=0) and accelerates at [math]3m/s^2[/math]  for 6 seconds. Find displacement.
  • [math]s = ut + \frac{1}{2} a t^2 \\
    s = (0 \times 6) + \frac{1}{2} (3) \times (6)^2 \\
    s = \frac{1}{2} (3) \times (6)^2 \\
    s = \frac{108}{2} \\
    s = 54 \, \text{m}[/math]

  • g) Motion with Uniform and Non-Uniform Acceleration

  • ⇒  Uniform Acceleration
  • A body experiences uniform acceleration when its velocity changes by the same amount in equal time intervals. The acceleration (a) remains constant.
  • ⇒  Equations (Uniform Acceleration)
  • For motion in a straight line with uniform acceleration, we use the kinematic equations:
  • [math]v = u + at \\
    s = ut + \frac{1}{2} a t^2 \\
    v^2 = u^2 + 2as[/math]
  • ⇒ Example:
  • A car accelerates uniformly from 0 to 30 m/s in 5 seconds. Find acceleration.
  • [math]a = \frac{v – u}{t} \\
    a = \frac{30 – 0}{5} \\
    a = 6 \, \text{m/s}^2[/math]
  • ⇒  Non-Uniform Acceleration
  • When acceleration varies over time, it is called non-uniform acceleration. This means velocity changes at different rates in equal time intervals.
  • – Example: A car moving in traffic speeds up and slows down randomly.
  • – Graphical Representation: The velocity-time graph for non-uniform acceleration is curved.
  • – Calculus Approach:
  • Instantaneous acceleration is given by
  • [math]a = \frac{dv}{dt}[/math]
  • Displacement is found using integration:
  • [math]s = \int v \, dt[/math]

  • h) Projectile Motion (Without Fluid Resistance)

  • A projectile is an object that moves under the influence of gravity after being launched. It follows a parabolic
  • 1. Motion Components
  • Projectile motion is 2D motion and is resolved into horizontal and vertical components:
Component Motion Type Equation Used
Horizontal (x-axis) Uniform motion (constant velocity) [math]v_x = u_x = u cos⁡θ, x = v_x t[/math]
Vertical (y-axis) Uniformly accelerated motion (gravity acts) [math]v_y = v_{y0} – g t \\
y = v_{y0} t – \frac{1}{2} g t^2 \\
v_y^2 = v_{y0}^2 – 2 g y[/math]
  • – Time of flight (T):
  • [math]T = \frac{2usinθ}{g}[/math]
  • – Maximum height (H):
  • [math]H = \frac{u^2 \sin^2 \theta}{2g}[/math]
  • – Range (R) [Horizontal Distance]:
  • [math]R = \frac{u^2 sin2θ}{g}[/math]
  • ⇒ Example:
  • A ball is launched at 20 m/s at 30°. Find time of flight.
  • [math]T = \frac{2u \sin \theta}{g} \\
    T = \frac{2(20) \sin(30^\circ)}{9.81} \\
    T = \frac{20}{9.81} \\
    T = 2.04 \, \text{s}[/math]

  • j)  Effect of Fluid Resistance on Projectiles

  • In reality, air resistance (drag) affects projectile motion. It reduces velocity and changes the trajectory.
  • ⇒  Qualitative Effects:
Without Fluid Resistance With Fluid Resistance
Symmetrical parabolic path Asymmetrical, shortened path
Velocity is only affected by gravity Velocity reduces due to air drag
Acceleration is constant (-9.81 m/s²) Acceleration decreases over time
Range is greater Range is shorter
Higher peak height Lower peak height
  • ⇒  Factor affecting fluid Resistance:
    1. Velocity – Higher speed → Greater resistance
    2. Shape – Aerodynamic shapes reduce drag
    3. Surface Area – Larger area increases drag
    4. Air Density – Denser air (e.g., humid conditions) increases resistance
  • Figure 5 Projectile motion
  • ⇒  Terminal Speed (Terminal Velocity)
  • When an object falls in fluid (air or water), it reaches terminal velocity when the drag force equals gravitational force:
  • [math]F_{\text{gravity}} = F_{\text{drag}}[/math]
  • ⇒ Example:
  • A skydiver initially accelerates but eventually reaches terminal velocity (~55 m/s).
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