# Question 6:

## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[P] = MLT^{-2}/L^2 = ML^{-1}T^{-2}$ | B1 | Penalise wrong dimensional symbols only once as accuracy mark. Penalise $+$ instead of $\times$ when combining dimensions only once as accuracy mark |

## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[\frac{1}{2}mu^2]$ or $[\frac{1}{2}mv^2]$ or $[W] = ML^2T^{-2}$ | M1 | $\frac{1}{2}$ not necessary |
| $[mP] = M^2L^{-1}T^{-2}$ so the equation is dimensionally inconsistent | A1 | Correct dimensions for $mP$ and conclusion |

## Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[\text{RHS}] = [M^0 L^{\alpha+\beta} T^{-(\alpha+\beta)}]^\gamma$ | M1 | e.g. RHS has no M |
| while LHS has $M^1$ so the equation must be dimensionally inconsistent | A1ft | or "some M" oe. Or "no M" compared to LHS. Ignore one minor slip in L or T. And $[W] = ML^2T^{-2}$ so comparing indices leads to a contradiction. Allow incorrect expression for $[W]$ from part (b), provided it includes an element of M |

## Part (d)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Because there are 4 unknowns and DA can only give us a maximum of 3 equations | B1 | Must be specific comparison. Or 3 equations in 4 unknowns seen (condone one slip), with appropriate comment about the equations in L and T at least. Must show all 3 equations, not just the two that involve $\alpha, \beta$ and $\delta$ |

## Part (d)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $ML^2T^{-2} = L^3T^{-3}L^\beta T^{-2\beta}M^\gamma T^\delta$ | B1ft | Correct dimensional expansion of both sides with $\alpha = 3$ substituted. $M^\gamma L^{3+\beta}T^{\delta-3-2\beta}$. Award if seen in part (i) (with $\alpha = 3$) |
| M: $1 = \gamma$, L: $2 = 3 + \beta$, T: $-2 = \delta - 3 - 2\beta$ | M1 | Correctly comparing indices for all three dimensions. Allow 1 slip. M0 if M does not appear on both sides of the dimensional equation |
| $\beta = -1,\ \gamma = 1,\ \delta = -1$ | A1 | |

## Part (d)(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| The resultant formula is $W = \frac{ku^3m}{at}$. This is unlikely to be correct since it suggests, for example, that the total work done becomes (very) small or negligible | B1 | Correct conclusion from their $\delta \leq 0$. Condone "not correct/incorrect". Answers referring to $t = 0$ are not valid for this question. Condone statements such as "…because work done decreases as time increases" oe, (which contradicts the expectation of a positive relationship between W and t) |

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