Distance in the Nth Second
For constant acceleration, displacement during the nth second of unit duration is s_n = u + (a/2)(2n-1). More generally, use x(n)-x(n-1), and distinguish signed displacement from distance if reversal occurs.
Why this shows up in the exam
Successive-second comparisons · Finding acceleration from interval data · Free-fall distance ratios
Learn the idea
Subtract cumulative displacements to isolate the displacement during one numbered second. The nth second is an interval, not the instant t = n. Its displacement is what the particle has accumulated by n seconds minus what it had accumulated by n-1 seconds.
🧠 Memory hook: Nth second means total at n minus total at n minus one.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- s_n = x(n) - x(n-1) — displacement during the nth one-second interval
- s_n = u + (a/2)(2n-1) — constant-acceleration nth-second displacement
How to approach it
- 1Write cumulative x(t)
- 2Evaluate at the two interval ends
- 3Check the velocity sign inside the interval
Common slip-ups that cost marks
- •Using x(n) as the nth-second distance
- •Forgetting the interval starts at n-1
- •Using signed displacement as distance across a reversal
🌟 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.