CAIE A-Level Mathematics AS 4.4.3 Constant Acceleration Questions

$a=2.08 \quad[2.07911 \ldots]$
From $m g \sin 12=m a$.

Marking guidance:

Allow exact (e.g. $a=g \sin 12$ ).

$v^{2}=5^{2}+2 a \times 60$
For use of $v^{2}=u^{2}+2 a s$ with u=5 and s=60.
Allow sign errors but a must be either $g \sin 12$ or $g \cos 12$ only.

Speed $=16.6 \mathrm{~ms}^{-1}[16.567861 \ldots]$
AWRT 16.6

Alternative Method for Q5(a)

For attempt at work energy equation
3 terms, dimensionally correct.
Allow sign errors;
allow sin/cos mix on PE term - condone m missing
from all terms. Must be a weight component.

$\frac{1}{2} m v^{2}=\frac{1}{2} m \times 5^{2}+m g \times 60 \sin 12$
Correct equation.
(for reference: $60 \sin 12=12.4747 \ldots$ )

Speed $=16.6 \mathrm{~ms}^{-1}[16.567861 \ldots]$
AWRT 16.6

$R=m g \cos 12$
Resolving correctly perpendicular to the plane.

$m g \sin 12-F=m a$

*M1

Use of Newton's second law, correct number of terms;
allow sign errors;

Marking guidance:

allow $\sin / \cos$ mix (must be a weight component).

For use of F=0.03 R to get equation in a (and m ) only
Where R is a component of weight only (dimensionally correct but allow $\sin / \cos \operatorname{mix})$.
$$[\Rightarrow a=1.79[1.78567 \ldots .]]$$$60=5 t+\frac{1}{2} a t^{2}$ and solve for t
Dependent on previous two M marks.

For use of $s=u t+\frac{1}{2} a t^{2}$ with s=60, u=5 and their a or other complete method to find positive value(s) of t.

Time $=5.86 \mathrm{~s}[5.86260 \ldots]$
AWRT 5.86 (from using a=1.79 or better).

AWRT 5.85 (from using a=1.8 ).

AWRT 5.87 (from correct working).


Alternative method for Question 5(b): Using energy

$R=m g \cos 12$
Resolving correctly perpendicular to the plane.

$\frac{1}{2} m v^{2}-\frac{1}{2} m \times 5^{2}=60 \times m g \sin 12-60 \times F$

(*M1)

Use of work-energy principle, correctly number of relevant terms;
allow sign errors;
allow sin/cos mix on PE term (must be a weight component).

For use of F=0.03 R to get equation in v (and m ) only
Where R is a component of weight only (dimensionally correct but allow sin/cos mix)
$$[\Rightarrow v=15.5[15.46870 \ldots .]] .$$$60=\frac{1}{2}(5+v) t$ and solve for t
Dependent on previous two M marks.

Use of $s=\frac{1}{2}(u+v) t$ with s=60, u=5 and their v or other complete method to find positive value(s) of t.

Time $=5.86 \mathrm{~s}[5.86260 \ldots]$