Mathematics · May / June 2023 · Paper 51 · Question 7
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As printed in the paper
The question as text
7. α Figure 4 A car of mass 1200 kg is towing a trailer of mass 600 kg up a straight road, as shown in Figure 4. The road is inclined at an angle α to the horizontal, where sin α = 1 12 The driving force produced by the engine of the car is 3000 N. –2 The car moves with acceleration 0.75 m s The non-gravitational resistance to motion of the car is modelled as a constant force of magnitude 2R newtons • the trailer is modelled as a constant force of magnitude R newtons • The car and the trailer are modelled as particles. The tow bar between the car and trailer is modelled as a light rod that is parallel to the direction of motion. Using the model,
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7(a)[4]
3000 1200 sin
g
600 sin
g
2
R
R
1800 0.75
−
α
−
α
−
−
=
From exact working
R =
60 *
(
)
T
−
600 sin
g
α
−
60
=
600 0.75
Trailer:
7(b)[3]
OR
(
)
(
)
3000 1200 sin
−
g
α
−
2 60
−
T
=
1200 0.75
Car:
(T could be replaced by ( T
−
) in either equation, leading to
T = 1000
−
, so tension is 1000 )
T =
1000(N)
Equation of motion
7(c)[4]
−
60
−
600 sin
g
α
=
600
a
(or – 600a)
11
a
= −
= −
0.9166
12
11
0
=
12
2
+
2
−
12 d
d =
78.5 , 79 (m)
Notes for question 7
(a)
Equation of motion for the whole system (or for car AND trailer with T eliminated) to give
an equation in R only.
M1
sinαdoes not need to be substituted
Correct number of terms, forces resolved where appropriate, condone sign errors and
sin/cos confusion.
A1
Correct equation with at most one error.
sinαdoes not need to be substituted
A1
Correct equation.
sinαdoes not need to be substituted
A1*
Reach the GIVEN answer with at least one intermediate line of working and must come
from exact working.
(b)
Equation of motion for the trailer or for the car. Correct number of terms, forces resolved
where appropriate, condone sign errors and sin/cos confusion.
M1
sinαdoes not need to be substituted but R = 60 does
A1
Correct unsimplified equation.
sinαdoes not need to be substituted
A1
Correct answer for T
(c)
Form an equation of motion for the trailer to find the new acceleration. Correct number of
terms, forces resolved where appropriate, condone sign errors and sin/cos confusion.
M1
sinαdoes not need to be substituted but R = 60 does
A1
Correct unsimplified equation.
sinαdoes not need to be substituted
M1
Complete method, with a calculated acceleration that is not g, to find the distance travelled.
A1
Cao 2 or 3sf Must be positive.
N.B. Allow a negative value of d and made positive for the distance.
Allow working in column vectors and penalise answers to (a) and (b)
in column vector form ONCE at the first time it occurs.
(
)
(
)
9
i
+
23
j
−−
2
i
+
j
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