Find velocity by integrating acceleration

A question is this type if and only if the acceleration vector a(t) is given along with an initial or boundary condition for velocity, and the task requires integrating to find the velocity vector v(t).

3 questions · Moderate -0.1

Sort by: Default | Easiest first | Hardest first

OCR MEI M1 2006 January Q5

6 marks Moderate -0.3

5 The acceleration of a particle of mass 4 kg is given by $\mathbf { a } = ( 9 \mathbf { i } - 4 t \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$, where $\mathbf { i }$ and $\mathbf { j }$ are unit vectors and $t$ is the time in seconds.

Find the acceleration of the particle when $t = 0$ and also when $t = 3$.
Calculate the force acting on the particle when $t = 3$. The particle has velocity $( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ when $t = 1$.
Find an expression for the velocity of the particle at time $t$.

Edexcel Paper 3 2020 October Q3

12 marks Standard +0.3

1. At time $t$ seconds, where $t \geqslant 0$, a particle $P$ moves so that its acceleration a $\mathrm { ms } ^ { - 2 }$ is given by

$$\mathbf { a } = ( 1 - 4 t ) \mathbf { i } + \left( 3 - t ^ { 2 } \right) \mathbf { j }$$ At the instant when $t = 0$, the velocity of $P$ is $36 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$

Find the velocity of $P$ when $t = 4$
Find the value of $t$ at the instant when $P$ is moving in a direction perpendicular to i
(ii) At time $t$ seconds, where $t \geqslant 0$, a particle $Q$ moves so that its position vector $\mathbf { r }$ metres, relative to a fixed origin $O$, is given by $$\mathbf { r } = \left( t ^ { 2 } - t \right) \mathbf { i } + 3 t \mathbf { j }$$ Find the value of $t$ at the instant when the speed of $Q$ is $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

AQA M2 2006 January Q5

8 marks Moderate -0.3

5 A particle moves such that at time $t$ seconds its acceleration is given by $$( 2 \cos t \mathbf { i } - 5 \sin t \mathbf { j } ) \mathrm { m } \mathrm {~s} ^ { - 2 }$$

The mass of the particle is 6 kg . Find the magnitude of the resultant force on the particle when $t = 0$.
When $t = 0$, the velocity of the particle is $( 2 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. Find an expression for the velocity of the particle at time $t$.