Pearson Edexcel IAL Mathematics M2.1.4 Differentiation & integration

Practise differentiating and integrating vector motion with respect to time to find position, velocity, acceleration, force and direction.

Exam points

differentiate r or v component by component to find velocity or acceleration vectors
integrate v with an initial position vector to find r at a given time
use vector direction conditions or |F|=m|a| to solve for time or force magnitude

[Maximum number: 7]

At time t seconds, t>0, a particle P is at the point with position vector r m, where

\mathbf{r}=\left(t^{4}-8 t^{2}\right) \mathbf{i}+\left(6 t^{2}-2 t^{\frac{3}{2}}\right) \mathbf{j}

(a)

Find the velocity of P when P is moving in a direction parallel to the vector j

[ 4 ]

Differentiate r to obtain v.

M1

$\mathbf v=(4t^3-16t)\mathbf i+(12t-3t^{1/2})\mathbf j$

A1

Set the i-component equal to zero and solve:

$4t^3-16t=0\Rightarrow t=2$

DM1

$\mathbf v=(24-3\sqrt2)\mathbf j\ \mathrm{m\,s^{-1}}$

A1

(b)

Find the acceleration of P when t=4

[ 3 ]

Differentiate v (both components)

M1

$\mathbf a=(12t^2-16)\mathbf i+\left(12-\frac32t^{-1/2}\right)\mathbf j$

A1

$\mathbf a(4)=176\mathbf i+\frac{45}{4}\mathbf j\ \mathrm{m\,s^{-2}}$

A1

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M2.1.4 - Differentiating and integrating vectors question 1

Question (a)

Question (b)