Impulse
- The force and momentum equation can be rearranged to find the impulse
- Impulse, I, is equal to the change in momentum:
I = FΔt = Δp = mv – mu
- Where:
- I = impulse (N s)
- F = force (N)
- t = time (s)
- p = momentum (kg m s–1)
- m = mass (kg)
- v = final velocity (m s–1)
- u = initial velocity (m s–1)
- This equation is only used when the force is constant
- Since the impulse is proportional to the force, it is also a vector
- The impulse is in the same direction as the force
- The unit of impulse is N s
- The impulse quantifies the effect of a force acting over a time interval
- This means a small force acting over a long time has the same effect as a large force acting over a short time
Rain vs Hail
- An example in everyday life of impulse is the effect of rain on an umbrella, compared to hail (frozen water droplets)
- When rain hits an umbrella, the water droplets tend to splatter and fall off it and there is only a very small change in momentum
- However, hailstones have a larger mass and tend to bounce back off the umbrella, creating a greater change in momentum
- Therefore, the impulse on an umbrella is greater in hail than in rain
- This means that more force is required to hold an umbrella upright in hail compared to rain
Since hailstones bounce back off an umbrella, compared to water droplets from rain, there is a greater impulse on an umbrella in hail than in rain
Worked example
A 58 g tennis ball moving horizontally to the left at a speed of 30 m s–1 is struck by a tennis racket which returns the ball back to the right at 20 m s–1.(a) Calculate the impulse delivered to the ball by the racket(b) State the direction of the impulse
Part (a)
Step 1: Write the known quantities
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-
- Taking the initial direction of the ball as positive (the left)
- Initial velocity, u = 30 m s–1
- Final velocity, v = –20 m s–1
- Mass, m = 58 g = 58 × 10–3 kg
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Step 2: Write down the impulse equation
Impulse I = Δp = m(v – u)
Step 3: Substitute in the values
I = (58 × 10–3) × (–20 – 30) = –2.9 N s
Part (b)
Direction of the impulse
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- Since the impulse is negative, it must be in the opposite direction to which the tennis ball was initial travelling (since the left is taken as positive)
- Therefore, the direction of the impulse is to the right
-
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 magnitude of 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 = 10 666 m s–1 = 11 km s–1
Step 5: State the final answer
- The final magnitude of the velocity of the ball is:
v = 11 km s–1
Examiner Tip
Remember that if an object changes direction, then this must be reflected by the change in sign of the velocity. As long as the magnitude is correct, the final sign for the impulse doesn't matter as long as it is consistent with which way you have considered positive (and negative)
For example, if the left is taken as positive and therefore the right as negative, an impulse of 20 N s to the right is equal to −20 N s
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
Areas
- 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!