Pearson Edexcel IAL Mathematics M1.4 Dynamics of a particle moving in a straight line or plane Question Bank

Practise applying Newton's laws, friction, impulse and momentum to particles moving in lines or planes with strings, slopes and collisions.

Exam points

resolve forces and apply F=ma to find acceleration, tension or resistance
use normal reaction and coefficient of friction to build equations on rough planes
apply impulse-momentum equations to find post-collision speed or impulse magnitude

Covered syllabus points

M1.4.1 - concept of a force M1.4.2 - Simple applications M1.4.3 - Momentum and impulse M1.4.4 - Coefficient of friction

[Maximum number: 3]

[In this question i and j are horizontal perpendicular unit vectors.]

A particle P of mass 2 kg moves on a smooth horizontal surface under the action of two forces $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ , where $\mathbf{F}_{1}=(-2 \mathbf{i}+3 \mathbf{j}) \mathrm{N}$ and $\mathbf{F}_{2}=(4 \mathbf{i}+2 \mathbf{j}) \mathrm{N}$ .

Find the acceleration of P.

At time t=0, the velocity of P is $(3 \mathbf{i}-4 \mathbf{j}) \mathrm{ms}^{-1}$

Resultant force
$=(-2 \mathbf{i}+3 \mathbf{j}+4 \mathbf{i}+2 \mathbf{j})=(2 \mathbf{i}+5 \mathbf{j})(\mathrm{N})$

B1

Seen or implied

Correct use of F=m a

M1

Must have added the forces

Acceleration
$=(\mathbf{i}+2.5 \mathbf{j})\left(\mathrm{ms}^{-2}\right)$

A1

Isw if they find the
magnitude

[3]

[Maximum number: 8]

Figure 3

A particle A of mass 4 kg is held at rest on a rough horizontal table. Particle A is attached to one end of a string that passes over a pulley P. The pulley is fixed the the of the table. The other end of the string is attached to a particle B, of mass 3 kg, which hangs freely below P.

The part of the string from A to P is perpendicular to the edge of the table and A, P and B all lie in the same vertical plane.

The string is modelled as being light and inextensible and the pulley is modelled as being small, smooth and light.

The system is released from rest with the string taut. At the instant of release, A is 2 m from the edge of the table and B is 1.4 m above a horizontal floor, as shown in Figure 3.

After descending with constant acceleration for 2 seconds, B hits the floor and does not rebound.

(a)

Show that the acceleration of A before B hits the floor is $0.7 \mathrm{~m} \mathrm{~s}^{-2}$

[ 2 ]

$1.4=\frac{1}{2} a \times 2^{2}$

M1

$a=0.7\left(\mathrm{~m} \mathrm{~s}^{-2}\right)^{*}$ GIVEN ANSWER

A1*

(2)

(b)

Find the value of $\mu$ .

[ 6 ]

$T-F=4 \times 0.7 \quad$ (for $A) \quad$ OR $\quad 3 g-F=7 \times 0.7 \quad$ (whole system)

M1 A1

$R=4 g ; F=\mu \times R$

B1; B1

$27.3-\mu \times 4 g=4 \times 0.7 \quad$ OR $\quad 3 g-\mu \times 4 g=7 \times 0.7$

DM1

$\mu=0.625$ or 0.63

A1

(6)

[Maximum number: 7]

A particle P has mass k m and a particle Q has mass m. The particles are movide direct each other in opposite directions along the same straight line when they collide directis

Immediately before the collision, P has speed 3 u and Q has speed u.
As a result of the collision, the direction of motion of each particle is reversed and the speed of each particle is halved.

(a)

Find the value of k.

[ 4 ]

Figure for Answer to Question (a) — Edexcel A-Level Mathematics AS

CLM: $k m \times 3 u-m u=-k m \times \frac{3}{2} u+m \times \frac{1}{2} u$

M1 A1 A1

$k=\frac{1}{3}$

A1

(4)

(b)

Find, in terms of m and u, the magnitude of the impulse exerted on Q in the collision.

[ 3 ]

$I=m\left(\frac{1}{2} u--u\right) \quad$ OR $\quad I=\frac{1}{3} m\left(\frac{3}{2} u--3 u\right)$

M1 A1

$I=\frac{3}{2} m u \quad$ must be positive

A1

(3)

(7)

Notes for question 2

All question bank results loaded

Covered syllabus points

M1.4 - Dynamics of a particle moving in a straight line or plane question 1

M1.4 - Dynamics of a particle moving in a straight line or plane question 2

Question (a)

Question (b)

M1.4 - Dynamics of a particle moving in a straight line or plane question 3

Question (a)

Question (b)