June 2025 Paper 3 Mechanics Q2
2.
A small box \(B\) of mass 2 kg is dragged in a straight line, along a rough horizontal plane, at a constant speed by a force of magnitude 5 N.
The line of action of the force makes an angle \(\alpha\) with the plane, where \(\sin\alpha = \dfrac{3}{5}\), as shown in Figure 2.
At the instant when \(B\) is at the point \(O\) on the plane, the force of magnitude 5 N is removed.
Given that after the force of magnitude 5 N is removed
- the box is modelled as a particle
- air resistance is modelled as being negligible
- the coefficient of friction between the box and the plane is modelled as 0.2
- the speed of the box as it passes through \(O\) is \(4\ \text{m s}^{-1}\)
- the box comes to rest at the point \(X\) on the plane
| Scheme | Marks | AO |
|---|---|---|
| Resolve vertically | M1 | 3.4 |
| \((\uparrow)\ 5\sin\alpha + R = 2g\) oe | A1 | 1.1b |
| \((R =)\ 19.6 - 3 = 16.6\) (N)* | A1* | 1.1b |
| (3) |
Notes
N.B. Penalise the use of \(g = 9.81\) ONCE (the first time it is used) for the whole question.
M1: Form an equation in \(R\) and \(\alpha\) only, with correct number of terms, condone sign errors and sin/cos confusion, \(\sin\alpha\) may or may not be substituted.
N.B. \(3 + R = 2g\) oe, with no evidence of resolving the 5 N, e.g. not using \(\sin\alpha\), is M0.
A1: Correct equation, \(\sin\alpha\) may or may not be be substituted.
A1*: Given answer correctly obtained, with at least one line of working.
| Scheme | Marks | AO |
|---|---|---|
| The box decelerates, slows down, loses speed or velocity, will come to a stop or rest, loses momentum | B1 | 2.4 |
| (1) |
Notes
B1: Any equivalent statement.
B0 if any incorrect extras e.g. box is stationary or at rest, doesn’t move, will stop moving.
| Scheme | Marks | AO |
|---|---|---|
| \((S =)\ 19.6\) or 20 (N) | B1 | 3.3 |
| (1) |
Notes
N.B. Penalise the use of \(g = 9.81\) ONCE (the first time it is used) for the whole question.
B1: Accept \(2g\) if \(g\) is not substituted for.
B0 if they use \(g = 9.81\) unless they have already been penalised in part (a).
| Scheme | Marks | AO |
|---|---|---|
| Equation of motion along the plane: \(-F = 2a\) or \(F = 2a\) | M1 | 3.4 |
| \(F = 0.2 \times\) their \(S\) | B1 | 1.1b |
| \(0^2 = 4^2 - 2 \times 0.2g \times d\) or \(0^2 = 4^2 + 2 \times 0.2g \times d\) or \(4^2 = 0^2 + 2 \times 0.2g \times d\) | M1 | 3.1b |
| \((OX =)\ 4.1\) (m) or 4.08 (m) | A1 | 1.1b |
| OR | ||
| Work done against friction \(= Fd\) | M1 | 3.4 |
| \(F = 0.2 \times\) their \(S\) | B1 | 1.1b |
| \(0.2 \times 2g \times d = \dfrac{1}{2} \times 2 \times 4^2\) | M1 | 3.1b |
| \((d = OX =)\ 4.1\) (m) or 4.08 (m) | A1 | 1.1b |
| (4) |
Notes
N.B. Penalise the use of \(g = 9.81\) ONCE (the first time it is used) for the whole question.
M1: Correct number of terms. Condone an extra \(g\) in \(ma\) term.
M0 if 2 is missing.
N.B. M0 if they use a vertical force for \(F\) e.g. \(2g\) or 16.6
B1: Seen (e.g. on a diagram) or implied.
M1: Complete method to form an equation in \(d\) (= \(OX\)) only, condone sign errors, using their calculated acceleration from an attempt at using \(F = ma\)
M0 if clearly using \(\mu\) (0.2) or \(-\mu\) for \(a\) without any calculation.
e.g. may find \(t\) first: \(0 = 4 - 0.2gt \Rightarrow t = \dfrac{20}{g}\ \left(= \dfrac{100}{49}\right)\)
then \(d = 4 \times \dfrac{20}{g} - \dfrac{1}{2} \times 0.2g \times \left(\dfrac{20}{g}\right)^2\)
or \(d = 0 - \dfrac{1}{2} \times (-0.2g) \times \left(\dfrac{20}{g}\right)^2\)
or \(0^2 = 4^2 + 2 \times (-0.2g) \times d\)
A1: Either answer. A0 for \(\dfrac{40}{g}\)
| Scheme | Marks | AO |
|---|---|---|
| the box (it) has been modelled as a particle the box (it) will have size or shape or dimensions the coefficient of friction has been modelled as being constant the coefficient of friction may not be exactly 0.2 or may vary the friction may vary B0: the ground may not be horizontal, an inaccurate value of \(g\) has been used, has not considered wind, the angle may not be accurate, any reference to the particle having mass (or not having mass). Ignore any reference to air resistance. | B1 | 3.5b |
| (1) | ||
| (10 marks) |
Notes
B1: Any equivalent statement which refers to the model. B0 if incorrect extras.