M11.3MechanicsStretch

Maxima and Minima Problems

Finding a maximum velocity or greatest height is an optimisation problem in disguise. The maximum or minimum of a quantity occurs where its rate of change is zero — so you differentiate and set the derivative to zero.

25 min Video by Zeeshan Zamurred Variable Acceleration

Edexcel Mechanics Y1 — Variable Acceleration playlist (Zeeshan Zamurred)Watch the full walkthrough before the notes below.

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What you'll be able to do

Find maximum/minimum velocity
Find greatest height or displacement
Use the condition: rate of change = 0
Interpret the result in context

The condition for a maximum

A quantity is at a maximum or minimum where its derivative (rate of change) is zero. So $maximum velocity$ occurs when $\frac{d v}{d t} = 0$ — that is, when the acceleration is zero.

max/min velocity \Rightarrow \frac{d v}{d t} = a = 0

Velocity is greatest when acceleration is zero.

1Acceleration

a = \frac{d v}{d t} = 12 - 6 t = 0

, so

t = 2

2Max velocity:

v = 12 (2) - 3 (2)^{2} = 24 - 12

Answer

12

m/s at

t = 2

Tip — Maximum velocity ⟺ acceleration = 0. Maximum displacement ⟺ velocity = 0.

Greatest displacement

The greatest displacement (e.g. maximum height) occurs when the $velocity$ is zero — the object stops moving in that direction before turning back. Set $v = 0$ , find $t$ , then substitute into $s$ .

Method summary

Differentiate the relevant quantity, set the derivative to zero to find the critical time, then substitute back to get the maximum or minimum value. State the answer with its time and units.

Formula recap

\frac{d v}{d t} = 0 \Rightarrow max/min velocity

Acceleration zero.

v = 0 \Rightarrow greatest displacement

Turning point of motion.

differentiate \to= 0 \to substitute back

The method.

Common mistakes to avoid

Setting velocity to zero to find maximum velocity.

Maximum velocity is where acceleration (dv/dt) = 0; v = 0 gives greatest displacement.

Finding the time but not the actual maximum value.

Substitute the critical time back into the quantity for its value.

Key takeaways

Max/min velocity: set acceleration (dv/dt) = 0.
Greatest displacement: set velocity (v) = 0.
Differentiate, set to zero, then substitute back for the value.

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