Moderate -0.5 This is a straightforward dimensional analysis check requiring students to verify each term has dimensions of power (ML²T⁻³). While it involves multiple terms and careful tracking of dimensions, it's a routine application of a standard technique with no problem-solving insight required—easier than average but not trivial due to the algebraic manipulation needed.
2 A car has mass \(m\) and travels up a slope which is inclined at an angle \(\theta\) to the horizontal. The car reaches a maximum speed \(v\) at a height \(h\) above its initial position. A constant resistance force \(R\) opposes the motion of the car, which has a maximum engine power output \(P\).
Neda finds a formula for \(P\) as
$$P = m g v \sin \theta + R v + \frac { 1 } { 2 } m v ^ { 3 } \frac { \sin \theta } { h }$$
where \(g\) is the acceleration due to gravity.
Given that the engine power output may be measured in newton metres per second, determine whether the formula is dimensionally consistent.
2 A car has mass $m$ and travels up a slope which is inclined at an angle $\theta$ to the horizontal. The car reaches a maximum speed $v$ at a height $h$ above its initial position. A constant resistance force $R$ opposes the motion of the car, which has a maximum engine power output $P$.
Neda finds a formula for $P$ as
$$P = m g v \sin \theta + R v + \frac { 1 } { 2 } m v ^ { 3 } \frac { \sin \theta } { h }$$
where $g$ is the acceleration due to gravity.\\
Given that the engine power output may be measured in newton metres per second, determine whether the formula is dimensionally consistent.
\hfill \mbox{\textit{AQA M3 2013 Q2 [6]}}