Distance–time & velocity–time graphs; acceleration

Distance–time graphs plot distance (y-axis) against time (x-axis), showing an object's journey. The gradient (slope) represents speed: steeper = faster. A horizontal line means the object is stationary (stopped). A straight diagonal line indicates constant speed, while a curve shows changing speed (acceleration or deceleration).

Velocity–time graphs plot velocity (y-axis) against time (x-axis). The gradient represents acceleration: positive slope = speeding up, negative slope = slowing down, horizontal line = constant velocity. The area under the graph equals displacement (distance travelled in a specific direction).

Key Definitions:

• Speed: distance travelled per unit time (scalar quantity). Formula: speed = distance ÷ time
• Velocity: speed in a specific direction (vector quantity)
• Acceleration: rate of change of velocity. Formula: a = (v - u) ÷ t where a = acceleration (m/s²), v = final velocity (m/s), u = initial velocity (m/s), t = time (s)
• Displacement: distance travelled in a specific direction from starting point

Essential Equations:

• Speed = distance ÷ time (v = s ÷ t)
• Acceleration = change in velocity ÷ time taken (a = Δv ÷ t)
• Average speed = total distance ÷ total time

Mnemonic: SUVAT for motion equations:

• S = displacement
• U = initial velocity
• V = final velocity
• A = acceleration
• T = time

Cambridge Note: Always include units in calculations: m/s for velocity, m/s² for acceleration, m for distance.

Understanding Distance–Time Graphs Through Real Life:

Imagine tracking a student cycling to school on your phone's GPS. A steep upward slope means they're cycling fast downhill, while a gentle slope indicates cycling slowly uphill. A horizontal line shows they've stopped at traffic lights. The journey isn't one straight line—it curves and changes, just like real movement.

Real-World Application: Police use distance–time analysis in accident investigations. Skid marks help reconstruct vehicle speeds before collisions. Formula rearrangement (distance = speed × time) determines if speed limits were exceeded.

Velocity–Time Graphs in Action:

Consider a sports car accelerating from traffic lights: the v-t graph shows a steep positive gradient (rapid acceleration). When cruising on the motorway, the graph becomes horizontal (constant velocity). Approaching the next junction, a negative gradient shows deceleration (braking).

Analogy: Think of velocity–time graphs like a financial graph. The gradient (slope) is your rate of saving money (acceleration = rate of velocity change). The total area under the line is your total savings (displacement = total distance travelled). A negative gradient means spending (deceleration).

Real-World Example: Roller coaster design relies heavily on v-t graphs. Engineers calculate maximum accelerations to ensure passenger safety (typically under 4g). The area under sections calculates distances needed for track layout.

Key Insight: Distance–time graphs answer "Where are you?"; velocity–time graphs answer "How fast are you going?" Understanding this distinction is crucial for Cambridge exams.

Example 1: Distance–Time Graph Analysis

Question: A cyclist travels 100m in 20s, stops for 10s, then travels 50m in 15s. Calculate: (a) speed in first section, (b) average speed for whole journey.

Solution: (a) Speed = distance ÷ time = 100m ÷ 20s = 5 m/s

Examiner Note: Always show working. Simply writing "5 m/s" loses method marks.

(b) Total distance = 100m + 50m = 150m Total time = 20s + 10s + 15s = 45s Average speed = 150m ÷ 45s = 3.33 m/s

Cambridge Tip: Average speed uses total distance and total time—include stationary periods.

Example 2: Velocity–Time Calculation

Question: A car accelerates from 10 m/s to 30 m/s in 5 seconds. Calculate: (a) acceleration, (b) distance travelled.

Solution: (a) Using a = (v - u) ÷ t a = (30 - 10) ÷ 5 = 20 ÷ 5 = 4 m/s²

(b) Method 1: Area under v-t graph = trapezium area Area = ½(a + b)h = ½(10 + 30) × 5 = ½ × 40 × 5 = 100m

Method 2: Using s = ½(u + v)t s = ½(10 + 30) × 5 = 100m

Mark Scheme Language: "Calculate" requires numerical answer with working. "Determine" or "Find" are equivalent commands.

Example 3: Graph Interpretation

Question: A velocity–time graph shows horizontal line at 15 m/s for 8s. Describe motion and calculate distance.

Solution: Horizontal line = zero gradient = zero acceleration = constant velocity of 15 m/s Distance = area = base × height = 8 × 15 = 120m