Impulse on a Force-Time Graph
- In real life, forces are often not constant and will vary over time
- If the force is plotted against time, the impulse is equal to the area under the force-time graph
When the force is not constant, the impulse is the area under a force–time graph
- This is because
Impulse = FΔt
- Where:
- F = force (N)
- Δt = change in time (s)
- The impulse is therefore equal whether there is
- A small force over a long period of time
- A large force over a small period of time
- The force-time graph may be a curve or a straight line
- If the graph is a curve, the area can be found by counting the squares underneath
- If the graph is made up of straight lines, split the graph into sections. The total area is the sum of the areas of each section
Worked example
A ball of mass 3.0 kg, initially at rest, is acted on by a force F which varies with t as shown by the graph.Calculate the velocity of the ball after 16 s.
Step 1: List the known quantities
- Mass, m = 3.0 kg
- Initial velocity, u = 0 m s-1 (since it is initially at rest)
Step 2: Calculate the impulse
- The impulse is the area under the graph
- The graph can be split up into two right-angled triangles with a base of 8 s and a height of 4 kN
Area = Impulse = 32 × 103 N s
Step 3: Write the equation for impulse
Impulse, I = Δp = m(v – u)
Step 4: Substitute in the values
I = mv
32 × 103 = 3.0 × v
v = (32 × 103) ÷ 3.0
v = 10666 m s–1 = 11 km s-1
Examiner Tip
Some maths tips for this section:Rate of Change
- ‘Rate of change’ describes how one variable changes with respect to another
- In maths, how fast something changes with time is represented as dividing by Δt (e.g. acceleration is the rate of change in velocity)
- More specifically, Δt is used for finite and quantifiable changes such as the difference in time between two events
- The area under a graph may be split up into different shapes, so make sure you’re comfortable with calculating the area of squares, rectangles, right-angled triangles and trapeziums!