Find mass or speed from a kinetic-energy value
Rearrange the squared-speed equation and compare controlled mass and speed changes.
Mapped to the named 0654 section using teaching from the corresponding single-science course. A 0654-specific review of every Core and Supplement objective is still in progress. Written practice is self-checked; it does not demonstrate supervised laboratory competence or award an official Cambridge grade.
Before you begin
- Calculate kinetic energy.
- Use a positive square root for a speed.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What non-negative number has square 36?
What you will learn
- Use m = 2Ek/v² and v = √(2Ek/m).
- Keep comparison controls and energy changes explicit.
Undo the square when finding speed
From Ek = ½mv², rearrange to v² = 2Ek/m and take the non-negative square root to find speed. For 36 J and 2 kg, v² = 36 and v = 6 m/s. Reporting 36 m/s omits the square-root step and confuses squared speed with speed. The energy alone does not supply a left or right direction.
To find mass, rearrange m = 2Ek/v². An object with 25 J at 5 m/s has mass 50/25 = 2.0 kg. A zero speed with zero energy does not determine a unique mass from 0/0; state when the given information is insufficient rather than carrying out an undefined quotient.
Comparisons need fixed quantities
At fixed speed, tripling mass triples kinetic energy. At fixed mass, tripling speed multiplies kinetic energy by nine. If mass doubles while speed halves, the combined factor is 2 × (½)² = ½. Do not use a speed-only rule when mass also changes, or add factors that belong to the multiplicative formula.
A 2 kg cart slowing from 6 to 3 m/s loses 27 J of kinetic energy, which can transfer to internal stores during braking. Its speed halves but its final energy is one quarter of its initial energy. A reversal at the same speed changes momentum but does not by itself change kinetic energy; the direction information still requires a separate observation.
Sources for this lesson
Pause and explain
Does 36 J at 2 kg determine a leftward or rightward velocity?
Worked example
An object of mass 2 kg has kinetic energy 36 J. Find its speed.
Show the worked solution
- Rearrange v² = 2Ek/m.
- v² = 2 × 36/2 = 36 m²/s².
- Speed v = √36 = 6 m/s; direction is not given by energy.
Answer 6 m/s; direction not determined by the energy.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find speed for 36 J kinetic energy and mass 2 kg.
Give me a hint
Find squared speed first, then its non-negative square root.
Compare my reasoning
- v² = 2 × 36/2 = 36.
- v = √36 = 6 m/s.
- No direction follows from these data alone.
6 m/s.
Look for these in your work
- Use the square root.
- Do not invent a direction.
Find mass for 25 J at 5 m/s and the energy factor if mass doubles while speed halves.
Give me a hint
Rearrange mass in the first task and multiply controlled factors in the second.
Compare my reasoning
- m = 2 × 25/5² = 2.0 kg.
- The mass factor is 2 and squared-speed factor is ¼.
- Combined energy factor = ½.
2.0 kg; energy becomes one half as large.
Look for these in your work
- Square the given speed.
- Include both changed controls.
A motionless object has zero kinetic energy. Can m = 2Ek/v² determine its mass from only Ek = 0 and v = 0? Explain.
Give me a hint
Inspect the denominator before evaluating the rearrangement.
Compare my reasoning
- The quotient would be 0/0.
- Any positive mass can have zero kinetic energy while stationary in this frame.
- Additional information is needed to determine the mass.
No; 0/0 is undefined and the supplied zero-energy rest state does not identify mass.
Look for these in your work
- Avoid an undefined calculation.
- Explain why multiple masses fit the stated data.
Common mistakes
- Take the square root for speed.
- Energy does not determine a velocity direction.
- Apply comparison factors only with their stated controls.
What can you explain now?
What energy factor follows if mass doubles and speed halves?
Compare with the explanation
One half.
The combined factor is 2 × (½)² = ½.
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Make it stick
Rearrange for mass and speed, state the square-root step and explain which controls a kinetic-energy ratio needs.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.