Use speed squared and distinguish energy from momentum.
Syllabus 4.5
Notes and key methods
A particle of mass M moving at speed v has kinetic energy equal to one half its mass times its speed squared, measured in joules. It is a scalar and does not become negative when the direction of motion reverses. Momentum instead uses a signed velocity without a square.
Doubling speed multiplies kinetic energy by four, while doubling mass at fixed speed multiplies it by two. To recover a speed from energy, rearrange first and then take the nonnegative square root. A change of energy is final minus initial.
\(K=\tfrac12 Mv^2\)
Check the conditions
Choose and retain a positive direction; distinguish signed quantities from magnitudes.
Use SI units and \(\displaystyle g= \frac{10m}{s^{2}}\) wherever gravity is relevant.
Apply only the model and time interval stated in the question.
Three worked examples
Example 1 · Kinetic energy and speed scaling
A particle has mass 4 kg and initially moves at speed \(\displaystyle \frac{10m}{s}\). Compare the stated separate speed changes without changing its mass.
Find its initial kinetic energy in J.
Find its kinetic energy if its speed is doubled, in J.
Find the increase in kinetic energy when its speed is doubled, in J.
Find the speed in m/s when its kinetic energy is nine times the initial value.
Show the reasoning
(a)
Use the complete kinetic-energy formula.
Half mass times speed squared gives \(200\) J.
Answer: \(200\).
(b)
Square the speed multiplier before applying it to energy.
The squared speed multiplier is \(\displaystyle 2^{2}\), so the new energy is \(800\) J.
Answer: \(800\).
(c)
Distinguish the new total energy from the increase.
Subtract initial energy from four times initial energy, giving \(600\) J.
Answer: \(600\).
(d)
Take a positive square root when reversing an energy multiplier.
At fixed mass, speed is multiplied by \(\displaystyle \sqrt{9}=3\), giving \(\displaystyle \frac{30m}{s}\).
Answer: \(30\).
Example 2 · Kinetic energy and speed scaling
A particle has mass 5 kg and initially moves at speed \(\displaystyle \frac{11m}{s}\). Compare the stated separate speed changes without changing its mass.
Find its initial kinetic energy in J.
Find its kinetic energy if its speed is doubled, in J.
Find the increase in kinetic energy when its speed is doubled, in J.
Find the speed in m/s when its kinetic energy is nine times the initial value.
Show the reasoning
(a)
Use the complete kinetic-energy formula.
Half mass times speed squared gives \(\displaystyle \frac{605}{2}\) J.
Answer: \(\displaystyle \frac{605}{2}\).
(b)
Square the speed multiplier before applying it to energy.
The squared speed multiplier is \(\displaystyle 2^{2}\), so the new energy is \(1210\) J.
Answer: \(1210\).
(c)
Distinguish the new total energy from the increase.
Subtract initial energy from four times initial energy, giving \(\displaystyle \frac{1815}{2}\) J.
Answer: \(\displaystyle \frac{1815}{2}\).
(d)
Take a positive square root when reversing an energy multiplier.
At fixed mass, speed is multiplied by \(\displaystyle \sqrt{9}=3\), giving \(\displaystyle \frac{33m}{s}\).
Answer: \(33\).
Example 3 · Kinetic energy and speed scaling
A particle has mass 6 kg and initially moves at speed \(\displaystyle \frac{12m}{s}\). Compare the stated separate speed changes without changing its mass.
Find its initial kinetic energy in J.
Find its kinetic energy if its speed is doubled, in J.
Find the increase in kinetic energy when its speed is doubled, in J.
Find the speed in m/s when its kinetic energy is nine times the initial value.
Show the reasoning
(a)
Use the complete kinetic-energy formula.
Half mass times speed squared gives \(432\) J.
Answer: \(432\).
(b)
Square the speed multiplier before applying it to energy.
The squared speed multiplier is \(\displaystyle 2^{2}\), so the new energy is \(1728\) J.
Answer: \(1728\).
(c)
Distinguish the new total energy from the increase.
Subtract initial energy from four times initial energy, giving \(1296\) J.
Answer: \(1296\).
(d)
Take a positive square root when reversing an energy multiplier.
At fixed mass, speed is multiplied by \(\displaystyle \sqrt{9}=3\), giving \(\displaystyle \frac{36m}{s}\).
Answer: \(36\).
Common mistakes
Do not omit the half factor or the speed square.
The energy increase on doubling speed is three times the old energy, not four times.