Ramiro and Lautaro are travelling from Buenos Aires to El Moro.

Ramiro travels in a vehicle whose velocity in ${\text{m}}{{\text{s}}^{ - 1}}$ is given by ${V_R} = 40 - {t^2}$, where $t$ is in seconds.

Lautaro travels in a vehicle whose displacement from Buenos Aires in metres is given by ${S_L} = 2{t^2} + 60$.

When $t = 0$, both vehicles are at the same point.

Find Ramiro’s displacement from Buenos Aires when $t = 10$.

METHOD 1

${S_L}(0) = 60$   (seen anywhere)     (A1)

recognizing need to integrate ${V_R}$     (M1)

eg     ${S_R}(t)\int {{V_R}{\text{d}}t}$

correct expression     A1A1

eg     $40t - \frac{1}{3}{t^3} + C$

Note:     Award A1 for $40t$, and A1 for $- \frac{1}{3}{t^3}$.

equate displacements to find C     (R1)

eg     $40(0) - \frac{1}{3}{(0)^3} + C = 60,{\text{ }}{S_L}(0) = {S_R}(0)$

$C = 60$     A1

attempt to find displacement     (M1)

eg     ${S_R}(10),{\text{ }}40(10) - \frac{1}{3}{(10)^3} + 60$

$126.666$

$126\frac{2}{3}{\text{ (exact), 127 (m)}}$     A1     N5

METHOD 2

recognizing need to integrate ${V_R}$     (M1)

eg     ${S_R}(t) = \int {{V_R}{\text{d}}t}$

valid approach involving a definite integral     (M1)

eg     $\int_a^b {{V_R}{\text{d}}t}$

correct expression with limits     (A1)

eg     $\int_0^{10} {\left( {40 - {t^2}} \right){\text{d}}t,{\text{ }}} \int_0^{10} {{V_R}{\text{d}}t,{\text{ }}\left[ {40t - \frac{1}{3}{t^3}} \right]} _0^{10}$

$66.6666$     A2

${S_L}(0) = 60$   (seen anywhere)     (A1)

valid approach to find total displacement     (M1)

eg     $60 + 66.666$

$126.666$

$126\frac{2}{3}$   (exact), $127$ (m)     A1     N5

METHOD 3

${S_L}(0) = 60$     (seen anywhere)     (A1)

recognizing need to integrate ${V_R}$     (M1)

eg     ${S_R}(t) = \int {{V_R}{\text{d}}t}$

correct expression     A1A1

eg     $40t - \frac{1}{3}{t^3} + C$

Note:     Award A1 for $40t$, and A1 for $- \frac{1}{3}{t^3}$.

correct expression for Ramiro displacement     A1

eg     ${S_R}(10) - {S_R}(0),{\text{ }}\left[ {40t - \frac{1}{3}{t^3} + C} \right]_0^{10}$

$66.6666$     A1

valid approach to find total displacement     (M1)

eg     $60 + 66.6666$

$126\frac{2}{3}$ (exact), 127 (m)     A1     N5

[8 marks]