Standard +0.8 This is a multi-step mechanics problem requiring students to set up two simultaneous equations from power-force-velocity relationships (P=Fv) in different scenarios, then solve for two unknowns. It involves understanding steady speed (zero acceleration, so driving force equals resistance plus component of weight on incline), applying P=Fv correctly in both cases, and solving the resulting system. While the individual concepts are standard M1 material, the problem requires careful setup, algebraic manipulation, and synthesis of multiple ideas, making it moderately challenging but not requiring novel insight.
A car has mass 1000 kg. When the car is travelling at a steady speed of \(v\) m s\(^{-1}\), where \(v > 2\), the resistance to motion of the car is \((Av + B)\) N, where \(A\) and \(B\) are constants. The car can travel along a horizontal road at a steady speed of 18 m s\(^{-1}\) when its engine is working at 36 kW. The car can travel up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.05\), at a steady speed of 12 m s\(^{-1}\) when its engine is working at 21 kW. Find \(A\) and \(B\). [7]
Solve two simultaneous equations in A and B only for A or B
Dependent on both previous M1’s
Answer
Marks
Guidance
A = 125, B = –250
A1
Both correct
7
Answer
Marks
Guidance
Question
Answer
Marks
Question 6:
6 | Case 1: DF = 36000/18
or
Case 2: DF = 21000/12 | B1 | DF = P/v in either case
18A + B = DF
[36000/18 = 18A + B = 2000] | M1 | Use DF = resistance (case 1)
18A + B = 2000 oe | A1 | Correct equation, unsimplified
12A + B = DF + weight component
[21000/12 = 12A + B + 1000 g × 1/20] | M1 | Use DF = resistance + weight component (case 2)
12A + B = 1250 oe | A1 | Correct equation, unsimplified
DM1 | Solve two simultaneous equations in A and B only for A or B
Dependent on both previous M1’s
A = 125, B = –250 | A1 | Both correct
7
Question | Answer | Marks | Guidance
A car has mass 1000 kg. When the car is travelling at a steady speed of $v$ m s$^{-1}$, where $v > 2$, the resistance to motion of the car is $(Av + B)$ N, where $A$ and $B$ are constants. The car can travel along a horizontal road at a steady speed of 18 m s$^{-1}$ when its engine is working at 36 kW. The car can travel up a hill inclined at an angle of $\theta$ to the horizontal, where $\sin \theta = 0.05$, at a steady speed of 12 m s$^{-1}$ when its engine is working at 21 kW. Find $A$ and $B$. [7]
\hfill \mbox{\textit{CAIE M1 2019 Q6 [7]}}