Edexcel IAL M1 January 2021 question paper and mark scheme (WME01/01)

Try 3 M1 questions like these

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1. Kinematics · 5 marks

   A particle moving with constant acceleration passes three points $A$, $B$ and $C$, in that order, on a straight line. The distance $AB$ is $20 \text{ m}$ and the distance $AC$ is $84 \text{ m}$. The speed of the particle at $B$ is $12 \text{ m s}^{-1}$ and its speed at $C$ is $20 \text{ m s}^{-1}$. Find the acceleration of the particle and its speed at $A$.

   Worked answer

   1. M1 for $BC = 84 - 20 = 64 \text{ m}$.
   2. M1 for $v^2 = u^2 + 2as$ from $B$ to $C$: $20^2 = 12^2 + 2a(64)$.
   3. A1 for $128a = 256 \Rightarrow a = 2 \text{ m s}^{-2}$.
   4. M1 for $v^2 = u^2 + 2as$ from $A$ to $B$: $12^2 = u_A^2 + 2(2)(20)$.
   5. A1 for $u_A^2 = 64 \Rightarrow u_A = 8 \text{ m s}^{-1}$.

2. Vectors (Mechanics context) · 8 marks

   Ship A starts at position $(0, 5)$ relative to a port and moves with a constant velocity vector of $\begin{pmatrix} 3 \\ -1 \end{pmatrix}\text{ km h}^{-1}$. Ship B starts at $(4, 0)$ and moves with a constant velocity vector of $\begin{pmatrix} 1 \\ 2 \end{pmatrix}\text{ km h}^{-1}$.

   1. Write down the position vectors of Ship A and Ship B at time $t$ hours.
   2. Find an expression for the square of the distance between the ships at time $t$, denoted $D^2(t)$.
   3. Find the exact time $t$ when the two ships are closest to each other.
   Worked answer

   1. M1 for writing the vector equations of motion.
      A1 for $\mathbf{r_A} = \begin{pmatrix} 3t \\ 5-t \end{pmatrix}$ and $\mathbf{r_B} = \begin{pmatrix} 4+t \\ 2t \end{pmatrix}$.
   2. M1 for finding the relative displacement vector $\mathbf{r_B} - \mathbf{r_A} = \begin{pmatrix} 4 - 2t \\ 3t - 5 \end{pmatrix}$.
      M1 for setting up the square of the magnitude: $D^2(t) = (4 - 2t)^2 + (3t - 5)^2$.
      A1 for expanding and simplifying to a quadratic: $D^2(t) = 16 - 16t + 4t^2 + 9t^2 - 30t + 25 = 13t^2 - 46t + 41$.
   3. M1 for recognizing that minimum distance occurs when the derivative of $D^2(t)$ is zero (or using the vertex formula $t = -b/2a$).
      M1 for solving $26t - 46 = 0 \implies 26t = 46$.
      A1 for finding exactly $t = \frac{23}{13}$ hours.

3. Momentum and Impulse · 4 marks

   A ball of mass $0.5 \text{ kg}$ is dropped from rest from a height of $19.6 \text{ m}$ above horizontal ground. It rebounds to a height of $4.9 \text{ m}$. Calculate the magnitude of the impulse exerted by the ground on the ball. (Take $g = 9.8 \text{ m s}^{-2}$)

   Worked answer

   1. M1 for finding the velocity just before impact using $v^2 = u^2 + 2as$: $v_1^2 = 0 + 2(9.8)(19.6) = 19.6^2 \implies v_1 = 19.6 \text{ m s}^{-1}$ downwards.
   2. M1 for finding the velocity just after impact using $v^2 = u^2 + 2as$: $0 = v_2^2 - 2(9.8)(4.9) \implies v_2^2 = 9.8^2 \implies v_2 = 9.8 \text{ m s}^{-1}$ upwards.
   3. M1 for calculating the impulse: $I = m(v_2 - v_1) = 0.5(9.8 - (-19.6))$.
   4. A1 for evaluating $I = 0.5(29.4) = 14.7 \text{ N s}$.

More on these topics: Kinematics · Vectors (Mechanics context) · Momentum and Impulse

M1 topics this paper can test

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• Kinematics
• Forces and Newton's Laws
• Moments
• Vectors (Mechanics context)
• Dynamics of a Particle
• Statics of a Particle
• Momentum and Impulse

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Formula booklet

Mathematical Formulae and Statistical Tables (International AS/A Level Mathematics) · Pearson Edexcel · For the 2018 IAL specification (current)