Find the initial velocity of P.

Find the maximum speed of P.

Write down the number of times that the acceleration of P is 0 m s⁻² .

Find the acceleration of P when it changes direction.

Find the total distance travelled by P.

initial velocity when t = 0      (M1)

eg v(0)

v = 17 (m s⁻¹)      A1 N2

[2 marks]

recognizing maximum speed when \(\left| v \right|\) is greatest      (M1)

eg  minimum, maximum, v' = 0

one correct coordinate for minimum      (A1)

eg  6.37896, −24.6571

24.7 (ms⁻¹)     A1 N2

[3 marks]

recognizing a = v ′     (M1)

eg  \(a = \frac{{{\text{d}}v}}{{{\text{d}}t}}\), correct derivative of first term

identifying when a = 0      (M1)

eg  turning points of v, t-intercepts of v ′

3       A1 N3

[3 marks]

recognizing P changes direction when v = 0       (M1)

t = 0.863851      (A1)

−9.24689

a = −9.25 (ms⁻²)      A2 N3

[4 marks]

correct substitution of limits or function into formula      (A1)
eg   \(\int_0^7 {\left| {\,v\,} \right|,\,\int_0^{0.8638} {v{\text{d}}t - \int_{0.8638}^7 {v{\text{d}}t} } ,\,\,\int {\left| {\,7\,{\text{cos}}\,x - 5{x^{{\text{cos}}\,x}}\,} \right|} \,dx,\,\,3.32 = 60.6} \)

63.8874

63.9 (metres)      A2 N3

[3 marks]