CAIE A-Level Mathematics AS 4.4.4 Connected Particles and Tow Systems

[Maximum number: 9]

A van of mass 4500 kg is towing a trailer of mass 750 kg down a straight hill inclined at an angle of $\theta$ to the horizontal where $\sin \theta=0.05$ . The van and the trailer are connected by a light rigid tow-bar which is parallel to the road. There are constant resistance forces of 2500 N on the van and 300 N on the trailer.

(a)

It is given that the tension in the tow-bar is 450 N .

Find the acceleration of the trailer and the driving force of the van's engine.

On another occasion, the van and trailer ascend a straight hill inclined at an angle of $\alpha$ to the horizontal where $\sin \alpha=0.09$ . The driving force of the van's engine is now 9100 N , and the speed of the van at the bottom of the hill is $20 \mathrm{~ms}^{-1}$ . The resistances to motion are unchanged.

[ 4 ]

Use of Newton's second law for van or trailer or system

Note: Trailer has 4 terms

Van has 5 terms

System has 7 terms (or 5 if counting van and trailer as one body)

M1*

Must have correct number of relevant terms. Allow sign errors.

Marking guidance:

Allow $\sin / \cos$ mix. Allow g missing.

Masses must be correct for their equation(s).

Forces must have components (or not) as required.

Must have either 0.05 or $\sin 2.86 \ldots$ or $\sin 2.9$ , not just $\sin \theta$ .

Trailer: $450+750 g \times 0.05-300=750 a$ [525=750 a]

Van: $D+4500 g \times 0.05-2500-450=4500 a$ [D-700=4500 a]

System:

$D+4500 g \times 0.05+750 g \times 0.05-2500-300=(4500+750) a$ [D-175=5250 a]
For any two correct equations.

For attempt to solve for a or D
Must get to ' a= ' or ' D= '. Must have correct number of relevant
terms in the equation(s) which they are using to find a or D . g
must be present. Allow sign errors. Allow sin/cos mix. If no working shown to solve their equations, then their answers should be correct for their equations.

$a=0.7 \mathrm{~ms}^{-2}$ and $D=3850 \mathrm{~N}$

(b)

Find the acceleration of the van and the tension in the tow-bar.

[ 5 ]

Use of Newton's second law for van or trailer or system

Note: Trailer has 4 terms

Van has 5 terms

System has 7 terms (or 5 if counting van and trailer as one body)

M1*

Must have correct number of relevant terms Allow sign errors. Allow sin/cos mix. Allow g missing.

Masses must be appropriate for their equation(s). Forces must have components (or not) as required. Must have either 0.09 or $\sin 5.16 \ldots$ or $\sin 5.2$ not just $\sin \theta$ .

Trailer: $T-300-750 g \times 0.09=750 a$ [T-975=750 a]
Van: $9100-2500-4500 g \times 0.09-T=4500 a$ [2550-T=4500 a]
System: $9100-2500-300-(4500+750) g \times 0.09=(4500+750) a$ [1575=5250 a]
A1 for one correct equation, second A1 for another correct equation.

If using Van and Trailer equations, must be using the same T for both to get the second A1.

For attempt to solve for a or T
Must get to ' a= ' or ' T= '. Must have correct number of relevant
terms in the equation(s) which they are using to find a or T . g
must be present. Allow sign errors. Allow sin/cos mix.
If no working shown to solve their equations, then their answers should be correct for their equations.

$T=1200 \mathrm{~N}$ and $a=0.3 \mathrm{~ms}^{-2}$

CAIE A-Level Mathematics AS 4.4.4 Connected Particle Models Questions

Exam points

CAIE A-Level Mathematics AS 4.4.4 Connected Particle Models Questions question 1

Question (a)

Question (b)