Velocity-Time Graphs
For v(t), acceleration is dv/dt and displacement is the signed integral integral(v dt). Distance is integral(|v| dt), so negative graph regions contribute positively to distance.
Why this shows up in the exam
Piecewise acceleration · Distance versus displacement comparisons · Catch-up problems represented graphically
Learn the idea
A velocity-time graph gives acceleration by slope and displacement by signed area. The graph's height says the current signed velocity. The area accumulated above the time axis moves position forward, while area below it moves position backward.
🧠 Memory hook: On v-t, slope is a and signed area is displacement.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- Delta x = integral(v dt) — signed area under a velocity-time curve
- distance = integral(|v| dt) — sum of absolute areas
- a = dv/dt — slope of the velocity-time graph
How to approach it
- 1Split the graph at corners and axis crossings
- 2Compute signed geometric areas
- 3Take absolute areas only for total distance
Common slip-ups that cost marks
- •Adding negative area as negative distance
- •Reading area as acceleration
- •Ignoring time-scale or velocity-scale units
🌟 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.