October 2020 Paper 1 Q9
9 A particle is moving in a straight line. The acceleration \(a\,\mathrm{m\,s^{-2}}\) of the particle at time \(t\) s is given by \(a = 0.8t + 0.5\). The initial velocity of the particle is \(3\,\mathrm{m\,s^{-1}}\) in the positive \(x\)-direction.
Determine whether the particle is ever stationary. [6]
| Scheme | Marks | AO |
|---|---|---|
| \(v = \displaystyle\int (0.8t + 0.5)\,\mathrm{d}t = 0.4t^2 + 0.5t + c\) | M1 | 3.1b |
| When \(t = 0,\ v = 3\) \(3 = 0.4 \times 0^2 + 0.5 \times 0 + c\) | M1 | 3.1b |
| So \(v = 0.4t^2 + 0.5t + 3\) | A1 | 1.1 |
| Particle stationary when \(v = 0\) \(0.4t^2 + 0.5t + 3 = 0\) | M1 | 3.1b |
| discriminant \(0.5^2 - 4 \times 0.4 \times 3 = -4.55 < 0\) | M1 | 3.1a |
| So the velocity is never zero and the particle never stationary. | E1 | 2.2a |
| [6] |
Notes
M1: Attempt to integrate, condone omission of +c
M1: Attempt to evaluate \(c\)
A1: Any form
M1: Forming an equation using their \(v = 0\)
M1: Use of discriminant or completing the square, showing equation has complex roots or stating that the equation has no real roots
Allow this M mark for solving their equation if it has real solutions
E1: Clear conclusion in context consistent with their working.
FT their \(v\). Dependent on at least 1 method mark.
Ignore any reference to \(t < 0\) but do not allow stationary at \(t = 0\)
(corrected from the printed mark scheme: the discriminant is printed as \(0.5^5 - 4 \times 0.4 \times 3\); it should be \(0.5^2\))
Alternative (OR)
| Scheme | Marks |
|---|---|
| \(v = \displaystyle\int (0.8t + 0.5)\,\mathrm{d}t = 0.4t^2 + 0.5t + c\) | M1 M1 |
| When \(t = 0,\ v = 3\) \(3 = 0.4 \times 0^2 + 0.5 \times 0 + c\) | A1 |
| So \(v = 0.4t^2 + 0.5t + 3\) | M1 |
| Clearly \(v\) is always positive therefore never sationary | M1 E1 |
M1: Attempt to integrate, condone omission of +c
M1: Attempt to evaluate \(c\)
M1: Uses the positivity of \(t\) to establish the positivity of \(v\).
M1: Argues that \(v\) is always positive
E1: Clear conclusion in context consistent with their working.
FT their \(v\). Dependent on at least 1 method mark.
Alternative (OR)
| Scheme | Marks |
|---|---|
| for \(t > 0\quad a = 0.8t + 0.5 > 0\) | M1 M1 A1 |
| So \(v\) is an increasing function | M1 |
| When \(t = 0,\ v = 3 > 0\) \(v[> 3] > 0\) for all values of \(t\) | M1 |
| So the velocity is never zero and the particle never stationary. | E1 |
M1: Attempt to construct an argument based on the positivity of \(v\).
M1: Uses the positivity of \(t\) aiming to establish the positivity of \(a\)
A1: Clear argument that \(a > 0\)
M1: Uses the link between \(a > 0\) and \(v\)
M1: Uses \(v_0 = 3\) explicitly in their argument
E1: Convincing complete argument.