5. A particle moves along the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the velocity of the particle is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing, where \(v = 2 t ^ { \frac { 3 } { 2 } } - 6 t + 2\)
At time \(t = 0\) the particle passes through the origin \(O\). At the instant when the acceleration of the particle is zero, the particle is at the point \(A\).
Find the distance \(O A\).
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Question 5:
Answer Marks
Guidance
Working Mark
Notes
Differentiate to find \(a\): \(a = \frac{dv}{dt} = 3t^{\frac{1}{2}} - 6\) M1
Powers going down
\(a = 3t^{\frac{1}{2}} - 6\) A1
Solve for \(a = 0\): M1
\(t^{\frac{1}{2}} = 2 \Rightarrow t = 4\) A1
Integrate to find \(s\): \(s = \int v \, dt\) M1
Powers going up
\(= \frac{4}{5}t^{\frac{5}{2}} - 3t^2 + 2t \, (+C)\) A1
Use limits 0 and 4: \(s = \frac{4}{5} \times 32 - 48 + 8 \, (= -14.4)\) DM1
Limits used correctly. Use of 0 can be implied. Dependent on preceding M1
Distance \(= 14.4\) m (14 m) A1
Or equivalent. Positive answer required
Total: [8] Question 6a:
Answer Marks
Guidance
Working Mark
Notes
Moments about \(A\): M1
Dimensionally correct. Condone sin/cos confusion
\(2.5N = 2\cos\theta \times 20\) A1
Correct unsimplified equation
\(N = \frac{2 \times \frac{4}{5} \times 20}{2.5} = 12.8\) (N) A1
Accept \(\frac{64}{5}\)
Total: (3) Question 6b:
Answer Marks
Guidance
Working Mark
Notes
Resolve \(\uparrow\): \(R + N\cos\theta + P\sin\theta = 20\) M1
1st equation. Dimensionally correct. Condone sin/cos confusion and sign errors
\((R = 9.76 - 0.6P)\) A1ft
Correct unsimplified equation in \(N\) or their \(N\)
Resolve \(\leftrightarrow\): \(F + P\cos\theta = N\sin\theta\) M1
2nd equation. Dimensionally correct. Condone sin/cos confusion and sign errors
\((F = 7.68 - 0.8P)\) A1ft
Correct unsimplified equation in \(N\) or their \(N\)
Use \(F = \frac{1}{4}R\) B1
Equation in \(P\) only: \(7.68 - 0.8P = \frac{1}{4}(9.76 - 0.6P)\), \((P = 8.06...)\) DM1
(or eliminate \(P\)). Dependent on preceding 2 M marks
Solve for \(\mu\): \(P = \mu N\) DM1
Dependent on preceding M mark
\(\mu = 0.630\), (0.63) A1
0.63 or better
Total: (8) Question 6b (Alternative):
Answer Marks
Guidance
Working Mark
Notes
Moments about \(C\): \(20 \times 0.5\cos\theta + F \times 2.5\sin\theta = R \times 2.5\cos\theta\) M1
1st equation. Dimensionally correct. Condone sin/cos confusion and sign errors
\((40 + 7.5F = 10R)\) A1
Correct unsimplified equation
Resolve parallel rod: \(P + F\cos\theta + R\sin\theta = 20\sin\theta \, (= 12)\) M1 A1
2nd equation. Correct unsimplified equation
Use \(F = \frac{1}{4}R\): \((8.125R = 40, \; R = 4.92...)\) B1
Solve for \(P\): \(P = 12 - \frac{R}{4} \times \frac{4}{5} - R \times \frac{3}{5} = 12 - \frac{4}{5}R = 8.06...\) DM1
Dependent on preceding 2 M marks
Solve for \(\mu\): \(P = \mu N\), \(\mu = 0.630\) (0.63) DM1 A1
Dependent on preceding M mark
Total: (8)
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# Question 5:
| Working | Mark | Notes |
|---------|------|-------|
| Differentiate to find $a$: $a = \frac{dv}{dt} = 3t^{\frac{1}{2}} - 6$ | M1 | Powers going down |
| $a = 3t^{\frac{1}{2}} - 6$ | A1 | |
| Solve for $a = 0$: | M1 | |
| $t^{\frac{1}{2}} = 2 \Rightarrow t = 4$ | A1 | |
| Integrate to find $s$: $s = \int v \, dt$ | M1 | Powers going up |
| $= \frac{4}{5}t^{\frac{5}{2}} - 3t^2 + 2t \, (+C)$ | A1 | |
| Use limits 0 and 4: $s = \frac{4}{5} \times 32 - 48 + 8 \, (= -14.4)$ | DM1 | Limits used correctly. Use of 0 can be implied. Dependent on preceding M1 |
| Distance $= 14.4$ m (14 m) | A1 | Or equivalent. Positive answer required |
**Total: [8]**
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# Question 6a:
| Working | Mark | Notes |
|---------|------|-------|
| Moments about $A$: | M1 | Dimensionally correct. Condone sin/cos confusion |
| $2.5N = 2\cos\theta \times 20$ | A1 | Correct unsimplified equation |
| $N = \frac{2 \times \frac{4}{5} \times 20}{2.5} = 12.8$ (N) | A1 | Accept $\frac{64}{5}$ |
**Total: (3)**
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# Question 6b:
| Working | Mark | Notes |
|---------|------|-------|
| Resolve $\uparrow$: $R + N\cos\theta + P\sin\theta = 20$ | M1 | 1st equation. Dimensionally correct. Condone sin/cos confusion and sign errors |
| $(R = 9.76 - 0.6P)$ | A1ft | Correct unsimplified equation in $N$ or their $N$ |
| Resolve $\leftrightarrow$: $F + P\cos\theta = N\sin\theta$ | M1 | 2nd equation. Dimensionally correct. Condone sin/cos confusion and sign errors |
| $(F = 7.68 - 0.8P)$ | A1ft | Correct unsimplified equation in $N$ or their $N$ |
| Use $F = \frac{1}{4}R$ | B1 | |
| Equation in $P$ only: $7.68 - 0.8P = \frac{1}{4}(9.76 - 0.6P)$, $(P = 8.06...)$ | DM1 | (or eliminate $P$). Dependent on preceding 2 M marks |
| Solve for $\mu$: $P = \mu N$ | DM1 | Dependent on preceding M mark |
| $\mu = 0.630$, (0.63) | A1 | 0.63 or better |
**Total: (8)**
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# Question 6b (Alternative):
| Working | Mark | Notes |
|---------|------|-------|
| Moments about $C$: $20 \times 0.5\cos\theta + F \times 2.5\sin\theta = R \times 2.5\cos\theta$ | M1 | 1st equation. Dimensionally correct. Condone sin/cos confusion and sign errors |
| $(40 + 7.5F = 10R)$ | A1 | Correct unsimplified equation |
| Resolve parallel rod: $P + F\cos\theta + R\sin\theta = 20\sin\theta \, (= 12)$ | M1 A1 | 2nd equation. Correct unsimplified equation |
| Use $F = \frac{1}{4}R$: $(8.125R = 40, \; R = 4.92...)$ | B1 | |
| Solve for $P$: $P = 12 - \frac{R}{4} \times \frac{4}{5} - R \times \frac{3}{5} = 12 - \frac{4}{5}R = 8.06...$ | DM1 | Dependent on preceding 2 M marks |
| Solve for $\mu$: $P = \mu N$, $\mu = 0.630$ (0.63) | DM1 A1 | Dependent on preceding M mark |
**Total: (8)**
---
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5. A particle moves along the $x$-axis. At time $t$ seconds, $t \geqslant 0$, the velocity of the particle is $v \mathrm {~ms} ^ { - 1 }$ in the direction of $x$ increasing, where $v = 2 t ^ { \frac { 3 } { 2 } } - 6 t + 2$
At time $t = 0$ the particle passes through the origin $O$. At the instant when the acceleration of the particle is zero, the particle is at the point $A$.
Find the distance $O A$.\\
(8)\\
\hfill \mbox{\textit{Edexcel M2 2019 Q5 [8]}}