Intuition What this page is forThe parent note gave you the formula K = 2 1 m v 2 Work–Energy Theorem W net = K f − K i not the same as knowing every trap it hides. This page marches through every kind of situation the topic can throw at you — positive and negative work, zero speed, doubling and tripling, a real crash problem, and an exam twist — so you never meet a case you haven't already seen solved.
Before we begin, three symbols must be crystal clear (we never use a symbol we haven't earned):
m mass in kilograms (kg) — how much "stuff" the object is.v speed in metres per second (m/s) — how fast, ignoring direction (that is why we can square it).K = 2 1 m v 2 kinetic energy in joules (J) — the motion-energy, always ≥ 0
Every problem in this topic lands in one of these cells . Our examples below are labelled by the cell they cover, so together they fill the whole grid.
Cell
Case class
What makes it tricky
Example
A Direct plug-in (m , v
none — sanity baseline
Ex 1
B Speed doubles / triples
K ∝ v 2 v Ex 2
C Speeding up → Δ K > 0
positive net work
Ex 3
D Slowing down → Δ K < 0
negative work (friction/braking)Ex 4
E Zero speed / degenerate input
K = 0 Ex 5
F Direction reverses (sign of v
v 2 Ex 6
G Real-world word problem
translate words → symbols; braking distance
Ex 7
H Exam twist: solve for a hidden variable
rearrange, don't just plug in
Ex 8
4 kg 5 m/s K Forecast: guess whether the answer is nearer 20 J or 50 J before reading.
Step 1. Write the definition: K = 2 1 m v 2 Why this step? The object starts nowhere and ends at a known speed — this is exactly what K
Step 2. Substitute m = 4 v = 5 K = 2 1 ( 4 ) ( 5 ) 2 = 2 1 ( 4 ) ( 25 ) = 50 J Why this step? Square the speed first , then take half of mass × that.
Verify: Units: kg ⋅ ( m/s ) 2 = kg⋅m 2 / s 2 = J
4 kg 15 m/s K Forecast: does energy triple, or grow more?
Step 1. Old energy (from Ex 1): K 1 = 50 J Why this step? We want a ratio , so we compare against a known baseline.
Step 2. New energy: K 2 = 2 1 ( 4 ) ( 15 ) 2 = 2 1 ( 4 ) ( 225 ) = 450 J Why this step? Plug the new speed straight into the definition.
Step 3. Ratio: K 1 K 2 = 50 450 = 9 Why this step? K ∝ v 2 3 2 = 9 nine-folds the energy.
Verify: 3 2 = 9 quadratic in speed — this is the single most important lesson of the topic.
0.5 kg 2 m/s 6 m/s Forecast: positive or negative? Bigger or smaller than 2 1 ( 0.5 ) ( 4 2 )
Step 1. Work–energy theorem: W net = K f − K i = 2 1 m ( v f 2 − v i 2 ) Why this step? We are asked about work , and the theorem links net work directly to the change in K
Step 2. Substitute:
W net = 2 1 ( 0.5 ) ( 6 2 − 2 2 ) = 2 1 ( 0.5 ) ( 36 − 4 ) = 2 1 ( 0.5 ) ( 32 ) = 8 J Why this step? We subtract the squares , not the speeds — direction of motion never enters.
Verify: K f = 2 1 ( 0.5 ) ( 36 ) = 9 J K i = 2 1 ( 0.5 ) ( 4 ) = 1 J 9 − 1 = 8 J added .
1200 kg 20 m/s 8 m/s Forecast: the sign must be _____ (fill in) — why?
Step 1. W net = 2 1 m ( v f 2 − v i 2 ) Why this step? Braking force is whatever it is; the theorem lets us skip it and use only speeds.
Step 2. Substitute:
W net = 2 1 ( 1200 ) ( 8 2 − 2 0 2 ) = 2 1 ( 1200 ) ( 64 − 400 ) = 600 × ( − 336 ) = − 201 , 600 J Why this step? v f < v i negative , dragging the whole answer negative.
Verify: ∣ W ∣ ≈ 2.0 × 1 0 5 J removed (turned to heat in the brake pads). Sign carries physical meaning ✓.
10 kg 3 m/s K Forecast: what is the kinetic energy of a thing that isn't moving?
Step 1. At rest v = 0 K i = 2 1 ( 10 ) ( 0 ) 2 = 0 J Why this step? The degenerate (zero-speed) case is the floor of kinetic energy — you cannot go lower, since v 2 ≥ 0
Step 2. Final: K f = 2 1 ( 10 ) ( 3 ) 2 = 2 1 ( 10 ) ( 9 ) = 45 J Why this step? Plug the final speed in.
Step 3. Net work = K f − K i = 45 − 0 = 45 J Why this step? Starting from rest, all the work becomes kinetic energy — this is the exact situation the original derivation assumed.
Verify: K ≥ 0 K i = 0 W net = K f
2 kg + 7 m/s − 7 m/s Forecast: velocity flipped sign — did the energy change?
Step 1. Before: K i = 2 1 ( 2 ) ( + 7 ) 2 = 2 1 ( 2 ) ( 49 ) = 49 J Why this step? The square of + 7 49
Step 2. After: K f = 2 1 ( 2 ) ( − 7 ) 2 = 2 1 ( 2 ) ( 49 ) = 49 J Why this step? The square of − 7 also 49
Step 3. Δ K = K f − K i = 49 − 49 = 0 J Why this step? Same speed means same kinetic energy, regardless of direction.
Verify: ( − 7 ) 2 = ( + 7 ) 2 = 49 p = m v did change (from + 14 − 14 Momentum vs Energy .
1000 kg 30 m/s 6000 N Forecast: all the motion-energy must be "eaten" by friction over the skid distance — guess: 50 m or 75 m?
Step 1. Kinetic energy to be removed: K = 2 1 ( 1000 ) ( 30 ) 2 = 2 1 ( 1000 ) ( 900 ) = 450 , 000 J Why this step? To stop the car, friction must do negative work equal in size to all of K
Step 2. Work by friction over distance s W = − F s = − 6000 s Why this step? Work = force × distance; friction points backward, so the sign is negative — this is the Work — definition and dot product with force and displacement anti-parallel.
Step 3. Work–energy theorem: W = K f − K i = 0 − 450 , 000 − 6000 s = − 450 , 000 ⇒ s = 6000 450 , 000 = 75 m . Why this step? Set the energy removed equal to the energy the car had, then solve for the unknown distance.
Verify: 6000 × 75 = 450 , 000 J K ∝ v 2 60 m/s 4 × 75 = 300 m
Worked example A pellet of mass0.02 kg 16 J Forecast: the answer is bigger than you expect — because we must undo a square (take a square root).
Step 1. Start from K = 2 1 m v 2 v v 2 = m 2 K ⇒ v = m 2 K . Why this step? v 2 1 m square-root — the inverse operation of squaring.
Step 2. Substitute K = 16 m = 0.02 v = 0.02 2 ( 16 ) = 0.02 32 = 1600 = 40 m/s . Why this step? Do the division inside first, then take the root.
Verify (plug back): 2 1 ( 0.02 ) ( 40 ) 2 = 2 1 ( 0.02 ) ( 1600 ) = 16 J
Recall Which cell am I in? (self-test)Given only m v K = 2 1 m v 2 W net = 2 1 m ( v f 2 − v i 2 ) K Δ K = 0 v 2 K m v v = 2 K / m F s = K s
Mnemonic The one rule that covers every cell"Squares, not sums." Every trap on this page comes from forgetting that energy lives on v 2
