KBAT Problems
KBAT Problem: Gradient and Area Together
A KBAT problem that needs both the gradient and the area under a graph in one question, connecting two graph skills.
Use each reading for its meaning
On a speed–time graph the gradient tells you the acceleration in one phase, and the area under a different phase tells you a distance. A KBAT question asks for both in one story, so you must know which reading answers which part.
Understand the problem
An original graph problem: a car's velocity–time graph rises in a straight line from 0 to 24 m/s over the first 8 seconds, then stays flat at 24 m/s until t = 20 s. One question asks for two things at once: the acceleration in the first phase and the total distance travelled.
On a velocity–time graph the gradient is acceleration and the area underneath is distance, so each reading is used for a different meaning.
Plan and solve
- Acceleration (first phase) = gradient = (24 − 0) ÷ (8 − 0) = 3 m/s².
- Distance in phase 1 = area of triangle = ½ × 8 × 24 = 96 m.
- Distance in phase 2 = area of rectangle = (20 − 8) × 24 = 12 × 24 = 288 m.
- Total distance = 96 + 288 = 384 m.
Check and a variant
Check: average speed = total distance ÷ total time = 384 ÷ 20 = 19.2 m/s, which sits between 0 and 24 m/s as it should. A twist: the examiner adds a braking phase from t = 20 s to t = 26 s, slowing to rest.
The braking gradient = (0 − 24) ÷ 6 = −4 m/s², and the extra distance = ½ × 6 × 24 = 72 m, giving a new total of 456 m. Same two skills, one more region.
Frequently asked questions
How do I know when to use gradient and when to use area?
Read the axes. On a velocity–time graph the gradient (steepness) is acceleration and the area under the line is distance.
On a distance–time graph the gradient is speed. Ask what units the gradient or area would carry; if they match what the question wants, that is the tool to use.
Why split the area into a triangle and a rectangle?
Because the shape under the graph is not a single standard figure, a sloping part then a flat part. Cutting it into a triangle and a rectangle lets you use area formulas you already know, then add the pieces.
For a curved graph you would estimate with the trapezium method instead.
Do units really matter in the working?
Yes. Acceleration must read m/s² and distance metres; writing a gradient without its unit, or mixing seconds and minutes, is a common way to lose marks even when the numbers are right.
Carrying units through every line also helps you catch when you have used the wrong reading for the wrong quantity.