A-Level Physics May/June 2025 Q1(b)(i): A constant resultant force F acts on a car of mass m. The car moves from rest with cons…

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  The work done $W$ on the car by the resultant force $F$ over a displacement $s$ is given by: $W = Fs$

  From Newton's second law, the resultant force is related to the mass $m$ and acceleration $a$ by: $F = ma$

  Substituting this into the work done equation gives: $W = (ma)s = mas$ [1]

  For an object moving with constant acceleration $a$ from rest ($u=0$), the final speed $v$ after a displacement $s$ is given by the equation of motion: $v^2 = u^2 + 2as$ $v^2 = 0 + 2as$ $v^2 = 2as$

  Rearranging this for the term $as$ gives: $as = \frac{v^2}{2}$ [2]

  Substitute this expression for $as$ into the equation for work done [1]: $W = m(\frac{v^2}{2})$ $W = \frac{1}{2}mv^2$

  By the work-energy principle, the work done on the car is equal to the gain in its kinetic energy, $E_k$. Therefore: $E_k = \frac{1}{2}mv^2$ [3]

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  How the marks are awarded

  • B1 — Stating the formula for work done ($W=Fs$) and combining it with Newton's second law ($F=ma$) to get the expression $W = mas$.
  • B1 — Correctly using the equation of motion $v^2 = u^2 + 2as$ with $u=0$ to derive the relationship $as = v^2/2$ (or an equivalent form).
  • B1 — Correctly substituting the kinematic relationship into the work done equation and equating work done with kinetic energy to complete the derivation and show $E_k = \frac{1}{2}mv^2$.

  Common mistakes

  • Starting the 'derivation' by stating $E_k = \frac{1}{2}mv^2$, which is circular reasoning as this is the formula that must be proven.
  • Forgetting to state the initial condition $u=0$ when using the equation of motion, leading to an incorrect relationship like $v^2 - u^2 = 2as$.
  • Successfully showing that $W = \frac{1}{2}mv^2$ but failing to make the final concluding statement that the work done is equal to the kinetic energy gained, i.e. $E_k = W$.
  • Attempting to use other equations of motion involving time ($t$), such as $s = \frac{1}{2}at^2$ and $v=at$, which requires more algebraic steps and introduces more opportunities for error.

  Examiner tip: For 'show that' questions, start from the most fundamental definitions (like Work Done = Force × Distance) and logically combine them with other known principles (like F=ma and suvat equations) step-by-step to reach the target equation.

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