Describing motion

• Position describes where an object is relative to an origin.

• Displacement is the change in position and is a vector: $<br>\Delta x = x_f - x_i$

• Distance is the total path length travelled and is a scalar.

• Speed is the rate of change of distance.

• Velocity is the rate of change of displacement and includes direction.

• Acceleration is the rate of change of velocity.

• Average velocity: $v_{avg} = \dfrac{\Delta x}{\Delta t}$

• Average speed: $\dfrac{\text{total distance}}{\text{total time}}$

• Instantaneous velocity/speed/acceleration are values at a particular moment.

• Sign convention matters: choose a positive direction first, then keep signs consistent.

This page supports the difference between distance, displacement, speed, and velocity. It is useful for checking why average speed and average velocity are not the same in round-trip motion. Source

Motion graphs and what they mean

• On a displacement–time graph, the gradient gives velocity.

• A straight-line displacement–time graph means constant velocity.

• A curved displacement–time graph means changing velocity.

• On a velocity–time graph, the gradient gives acceleration.

• The area under a velocity–time graph gives displacement.

• On an acceleration–time graph, the area under the graph gives change in velocity.

• Instantaneous values come from the gradient at a point.

• In exams, always state what the gradient/area represents before calculating.

This graph is a simple velocity–time graph used to visualize how gradient and area are interpreted in kinematics. It is most useful when revising how a graph can show changing motion without needing a full paragraph of explanation. Source

Uniformly accelerated motion

• These equations apply only when acceleration is constant.

• Use SUVAT carefully: $s, u, v, a, t$.

• $s = \dfrac{u+v}{2}t$

• $v = u + at$

• $s = ut + \dfrac{1}{2}at^2$

• $v^2 = u^2 + 2as$

• Pick the equation that avoids the variable you do not know.

• Near Earth’s surface, for vertical motion, use $a = \pm g$ depending on your sign convention.

• If acceleration is non-uniform, these constant-acceleration equations cannot be used directly over the whole motion.

These figures show the graphical meaning of the core kinematics relationships: slope of displacement–time = velocity and slope of velocity–time = acceleration. They are useful for linking equations to graph interpretation, which is heavily tested in IB-style questions. Source

Projectile motion without fluid resistance

• Treat projectile motion as independent horizontal and vertical motion.

• Horizontal motion: constant velocity because $a_x = 0$.

• Vertical motion: constant downward acceleration because $a_y = -g$ if upward is positive.

• Resolve launch velocity into components: $u_x = u\cos\theta$ and $u_y = u\sin\theta$.

• Apply kinematics separately in each direction, then combine results if needed.

• Time of flight usually comes from vertical motion.

• Range usually comes from horizontal motion: $\text{range} = v_x \times t$

• At the highest point, vertical velocity is $0$, but horizontal velocity is unchanged.

• In the absence of fluid resistance, the trajectory is parabolic.

• Be familiar with projectiles launched horizontally, above the horizontal, and below the horizontal.

This diagram shows the horizontal and vertical components of initial velocity for a projectile. It is ideal for revising how to split motion into two independent directions before using the kinematics equations. Source

This illustration shows a parabolic projectile path with launch angle, initial speed, maximum height, and range labeled. It is a strong summary image for the geometry of standard projectile questions. Source

This figure compares an object in free fall with one in projectile motion and shows that their vertical motion matches when air resistance is ignored. It is excellent for understanding the independence of horizontal and vertical motion. Source

Effect of fluid resistance

• With fluid resistance, projectile motion is more realistic but harder to model exactly.

• Fluid resistance acts opposite to the direction of motion.

• The path is no longer perfectly parabolic.

• Compared with no air resistance, time of flight, range, and often maximum height are usually reduced.

• Velocity changes in both magnitude and direction in a more complex way.

• Acceleration is no longer constant because it depends on both gravity and resistive force.

• A falling object can reach terminal speed when resistive force = weight, so resultant force = 0 and acceleration = 0.

• IB expects qualitative comparisons here: describe how trajectory, range, time of flight, velocity, acceleration, and terminal speed are affected.

These diagrams show how increasing air resistance reduces the net force and therefore the acceleration of a falling object. They help explain why an object eventually reaches terminal velocity when forces balance. Source