6. A particle \(P\) is moving with constant acceleration.
At time \(t = 1\) second, \(P\) has velocity \(( - \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
At time \(t = 4\) seconds, \(P\) has velocity \(( 5 \mathbf { i } - 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)
Find the speed of \(P\) at time \(t = 3.5\) seconds.
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Question 6:
Answer Marks
Guidance
Answer/Working Marks
Guidance
\((5\mathbf{i} - 8\mathbf{j}) = (-\mathbf{i} + 4\mathbf{j}) + 3\mathbf{a}\) M1 A1
Use of \(\mathbf{v} = \mathbf{u} + \mathbf{a}t\) with \(t=3\) oe; to give equation in a only; allow u and v reversed
\(\mathbf{v} = (-\mathbf{i} + 4\mathbf{j}) + 2.5(2\mathbf{i} - 4\mathbf{j})\) M1
Equation in v only using their a and \(t = 2.5\) oe; M0 if \(\mathbf{u} = \mathbf{0}\) assumed
\(\mathbf{v} = (4\mathbf{i} - 6\mathbf{j})\) A1
Speed \(= \sqrt{4^2 + (-6)^2} = \sqrt{52} = 7.2 \ (\text{ms}^{-1})\) or better M1 A1
Use of Pythagoras including square root; cao
OR (Ratio method):
Answer Marks
Guidance
Answer/Working Marks
Guidance
\((4-3.5)[\mathbf{v} - (-\mathbf{i}+4\mathbf{j})] = (3.5-1)[(5\mathbf{i}-8\mathbf{j}) - \mathbf{v}]\) oe M2 A1
M2 for equation in v only with correct structure; A1 correct unsimplified ratio equation
\(\mathbf{v} = (4\mathbf{i} - 6\mathbf{j})\) A1
Speed \(= \sqrt{4^2 + (-6)^2} = \sqrt{52} = 7.2 \ (\text{ms}^{-1})\) or better M1 A1
cao
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## Question 6:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(5\mathbf{i} - 8\mathbf{j}) = (-\mathbf{i} + 4\mathbf{j}) + 3\mathbf{a}$ | M1 A1 | Use of $\mathbf{v} = \mathbf{u} + \mathbf{a}t$ with $t=3$ oe; to give equation in **a** only; allow **u** and **v** reversed |
| $\mathbf{v} = (-\mathbf{i} + 4\mathbf{j}) + 2.5(2\mathbf{i} - 4\mathbf{j})$ | M1 | Equation in **v** only using their **a** and $t = 2.5$ oe; M0 if $\mathbf{u} = \mathbf{0}$ assumed |
| $\mathbf{v} = (4\mathbf{i} - 6\mathbf{j})$ | A1 | |
| Speed $= \sqrt{4^2 + (-6)^2} = \sqrt{52} = 7.2 \ (\text{ms}^{-1})$ or better | M1 A1 | Use of Pythagoras including square root; cao |
**OR (Ratio method):**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(4-3.5)[\mathbf{v} - (-\mathbf{i}+4\mathbf{j})] = (3.5-1)[(5\mathbf{i}-8\mathbf{j}) - \mathbf{v}]$ oe | M2 A1 | M2 for equation in **v** only with correct structure; A1 correct unsimplified ratio equation |
| $\mathbf{v} = (4\mathbf{i} - 6\mathbf{j})$ | A1 | |
| Speed $= \sqrt{4^2 + (-6)^2} = \sqrt{52} = 7.2 \ (\text{ms}^{-1})$ or better | M1 A1 | cao |
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6. A particle $P$ is moving with constant acceleration.
At time $t = 1$ second, $P$ has velocity $( - \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$\\
At time $t = 4$ seconds, $P$ has velocity $( 5 \mathbf { i } - 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$\\
Find the speed of $P$ at time $t = 3.5$ seconds.
\hfill \mbox{\textit{Edexcel M1 2022 Q6 [6]}}