5.3 Real-Life Graphs
Why Study Real-Life Graphs?
Graphs help us visualise and analyse real-world situations. In this lesson, we'll focus on graphs involving motion - how objects move through space and time.
Graphs help us visualise and analyse real-world situations. In this lesson, we'll focus on graphs involving motion - how objects move through space and time.
⚡ Two Important Graph Types:
1. Distance-Time Graphs:
• Show how far something has travelled over time
• Gradient = Speed
2. Velocity-Time Graphs:
• Show how fast something is moving over time
• Gradient = Acceleration
• Area under graph = Distance
• Show how far something has travelled over time
• Gradient = Speed
2. Velocity-Time Graphs:
• Show how fast something is moving over time
• Gradient = Acceleration
• Area under graph = Distance
Reading Distance-Time Graphs
On a distance-time graph, the y-axis shows distance and the x-axis shows time.
On a distance-time graph, the y-axis shows distance and the x-axis shows time.
⚡ What the Graph Shape Tells Us:
Horizontal line → Stationary (not moving)
Straight line going up → Moving at constant speed
Steeper line → Faster speed
Curve → Changing speed (accelerating or decelerating)
Line going down → Returning towards the start
Straight line going up → Moving at constant speed
Steeper line → Faster speed
Curve → Changing speed (accelerating or decelerating)
Line going down → Returning towards the start
Distance-Time Graph
Click Play to see the journey!
💡 Calculating Speed from a Distance-Time Graph:
The gradient (slope) of a distance-time graph gives the speed!
$\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \text{Gradient}$
The gradient (slope) of a distance-time graph gives the speed!
📖 Example: Sarah's Journey
Sarah walks to the shop and back home. Analyse her journey:
Section A (0-10 min): Walks 400m in 10 min
Speed = $\frac{400}{10}$ = 40 m/min
Section B (10-15 min): Horizontal line - at the shop (stationary)
Section C (15-20 min): Returns 400m in 5 min
Speed = $\frac{400}{5}$ = 80 m/min (faster on the way back!)
Speed = $\frac{400}{10}$ = 40 m/min
Section B (10-15 min): Horizontal line - at the shop (stationary)
Section C (15-20 min): Returns 400m in 5 min
Speed = $\frac{400}{5}$ = 80 m/min (faster on the way back!)
🎯 Calculate Speed from Graph
m/s
m/s
🎯 What's happening?
Reading Velocity-Time Graphs
On a velocity-time graph, the y-axis shows velocity (speed in a certain direction) and the x-axis shows time.
On a velocity-time graph, the y-axis shows velocity (speed in a certain direction) and the x-axis shows time.
⚡ What the Graph Shape Tells Us:
Horizontal line → Constant velocity (no acceleration)
Line going up → Accelerating (speeding up)
Line going down → Decelerating (slowing down)
Line at y = 0 → Stationary
Below x-axis → Moving in reverse direction
Line going up → Accelerating (speeding up)
Line going down → Decelerating (slowing down)
Line at y = 0 → Stationary
Below x-axis → Moving in reverse direction
Remember: On a distance-time graph, a horizontal line means stationary. On a velocity-time graph, a horizontal line means constant velocity.
Velocity-Time Graph
Click Play to see the motion!
💡 Calculating Acceleration:
The gradient of a velocity-time graph gives the acceleration!
• Positive gradient = speeding up
• Negative gradient = slowing down (deceleration)
$\text{Acceleration} = \frac{\text{Change in Velocity}}{\text{Time}} = \text{Gradient}$
The gradient of a velocity-time graph gives the acceleration!
• Positive gradient = speeding up
• Negative gradient = slowing down (deceleration)
📖 Example: A Car's Journey
Section A (0-4s): Accelerates from 0 to 20 m/s
Acceleration = $\frac{20-0}{4}$ = 5 m/s²
Section B (4-8s): Constant velocity at 20 m/s
Acceleration = 0
Section C (8-12s): Decelerates from 20 to 0 m/s
Acceleration = $\frac{0-20}{4}$ = -5 m/s² (negative = deceleration)
Acceleration = $\frac{20-0}{4}$ = 5 m/s²
Section B (4-8s): Constant velocity at 20 m/s
Acceleration = 0
Section C (8-12s): Decelerates from 20 to 0 m/s
Acceleration = $\frac{0-20}{4}$ = -5 m/s² (negative = deceleration)
🎯 Calculate Acceleration
m/s²
m/s²
🎯 Velocity-Time Graph Interpretation
Finding Distance from Velocity-Time Graphs
The area under a velocity-time graph gives the total distance travelled.
The area under a velocity-time graph gives the total distance travelled.
Area = Distance
Area = 15 × 6 = 90 m
📖 Example: Finding Total Distance
Triangle (0-4s): Area = ½ × 4 × 20 = 40 m
Rectangle (4-8s): Area = 4 × 20 = 80 m
Triangle (8-12s): Area = ½ × 4 × 20 = 40 m
Total Distance = 40 + 80 + 40 = 160 m
Rectangle (4-8s): Area = 4 × 20 = 80 m
Triangle (8-12s): Area = ½ × 4 × 20 = 40 m
Total Distance = 40 + 80 + 40 = 160 m
🎯 Calculate Distance from Area
m
m
Understanding Rates
A rate tells us how one quantity changes compared to another. The gradient of any graph gives the rate of change.
A rate tells us how one quantity changes compared to another. The gradient of any graph gives the rate of change.
⚡ Common Rates in Real Life:
Speed = Distance ÷ Time (m/s, km/h, mph)
Acceleration = Velocity ÷ Time (m/s²)
Flow rate = Volume ÷ Time (litres/minute)
Growth rate = Population change ÷ Time
Cost rate = Cost ÷ Quantity (£/kg)
Acceleration = Velocity ÷ Time (m/s²)
Flow rate = Volume ÷ Time (litres/minute)
Growth rate = Population change ÷ Time
Cost rate = Cost ÷ Quantity (£/kg)
📊 Water Tank Example
Rate of filling: 5 litres/minute
Example: A tank fills at a constant rate. After 4 minutes it contains 20 litres, and after 10 minutes it contains 50 litres. Find the rate of filling.
Rate = $\frac{\text{Change in volume}}{\text{Change in time}}$
Rate = $\frac{50 - 20}{10 - 4} = \frac{30}{6} = 5$ litres/minute
Rate = $\frac{50 - 20}{10 - 4} = \frac{30}{6} = 5$ litres/minute
💡 Estimating Gradient of a Curve:
For curved graphs, we can estimate the gradient at a point by drawing a tangent - a straight line that touches the curve at that point.
For curved graphs, we can estimate the gradient at a point by drawing a tangent - a straight line that touches the curve at that point.
1. Draw a tangent line at the point
2. Choose two points on the tangent
3. Calculate the gradient = (Difference in y)/(Difference in x)
2. Choose two points on the tangent
3. Calculate the gradient = (Difference in y)/(Difference in x)
Tangent Tool
Gradient at x = 5: approximately 10
🎯 Calculate Rate of Change
Side-by-Side Comparison
Understanding the relationship between these two graph types is important.
Understanding the relationship between these two graph types is important.
Comparison between Distance and Velocity Time graphs
Distance-Time
Velocity-Time
Click Play to compare!
⚡ Summary Table:
| Feature | Distance-Time | Velocity-Time |
|---|---|---|
| Gradient gives | Speed | Acceleration |
| Horizontal line means | Stationary | Constant velocity |
| Steeper line means | Faster | Greater acceleration |
| Area under graph gives | — | Distance |
🎯 Some Practice
Real Life Applications:
• Sat Nav: Shows distance and time remaining
• Speedometers: Display instantaneous velocity
• Athletics: Analysing sprinters' acceleration
• Car safety: Calculating stopping distances
• Roller coasters: Designing safe acceleration limits
• Sat Nav: Shows distance and time remaining
• Speedometers: Display instantaneous velocity
• Athletics: Analysing sprinters' acceleration
• Car safety: Calculating stopping distances
• Roller coasters: Designing safe acceleration limits