Differential and Integral Kinematics
For sufficiently smooth one-dimensional motion, v = dx/dt and a = dv/dt = d²x/dt². Conversely, integrating acceleration and velocity determines v and x up to constants fixed by initial conditions.
Why this shows up in the exam
Polynomial motion laws · Implicit x-t relations · Velocity laws given as functions of time
Learn the idea
Differentiate position to descend to velocity and acceleration; integrate to climb back with initial conditions. Position, velocity, and acceleration are three views of the same motion. Differentiation zooms into rates of change, while integration rebuilds accumulated change and needs a starting value.
🧠 Memory hook: Differentiate down x to v to a; integrate up with starting data.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- v = dx/dt — velocity from position
- a = dv/dt = d²x/dt² — acceleration from velocity or position
- x(t) = x0 + integral(v dt) — position from velocity and an initial position
How to approach it
- 1Identify the independent variable
- 2Differentiate or integrate exactly once per kinematic level
- 3Apply initial conditions before numerical substitution
Common slip-ups that cost marks
- •Dropping integration constants
- •Confusing distance with signed position
- •Differentiating an implicit relation as if variables were independent
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A particle has initial speed 2 m/s and constant acceleration 3 m/s^2 for 4 s. What distance does it cover?
More from Kinematics
Distance, Displacement, Speed, and Velocity
Understand the differences between distance and displacement, and between speed and velocity, including how to calculate average speed and average velocity in various scenarios.
Equations of Motion and Uniform Acceleration
Learn the kinematic equations for uniformly accelerated motion, including applications to free fall, retardation, and calculation of distance in specific time intervals.
Projectile Motion
Study the motion of projectiles, including the independence of horizontal and vertical components, trajectory equations, maximum height, and the effect of initial conditions.
Relative Velocity and Motion Analysis
Explore how to determine the velocity of one object relative to another and analyze motion from different reference frames, including periodic motion.
Position, Path Length, and Displacement
For one-dimensional motion, displacement over an interval is Delta x = x_f - x_i and may be positive, negative, or zero. Distance is the non-negative path length, so distance is always at least |Delta x|.
Speed and Velocity
Instantaneous velocity is the signed rate v = dx/dt, while instantaneous speed is |v|. In one dimension the sign of v identifies motion along or opposite the positive axis.