Challenging +1.2 This is a standard projectiles problem requiring resolution of velocity components and use of the perpendicularity condition. While it involves setting up equations for velocity at two times and applying the dot product condition, the method is straightforward once the setup is recognized. The 5-mark allocation and single-answer nature make it moderately challenging but not exceptional for Further Maths.
At time \(t = 0\) seconds, a particle \(P\) is projected with speed \(u\) m s\(^{-1}\) at an angle \(60°\) above the horizontal from a point \(O\). In the subsequent motion \(P\) moves freely under gravity. The direction of motion of \(P\) when \(t = 5\) is perpendicular to its direction of motion when \(t = 15\).
Find the value of \(u\). [5]
usin605g, and ucos60 , or usin6015g , and ucos60
B1
usin605g
If is direction of velocity at t 5, tan
Answer
Marks
Guidance
ucos60
M1*
Accept equivalent for t15.
usin605g usin6015g
For perpendicular directions, 1
Answer
Marks
Guidance
ucos60 ucos60
M1dep
Multiply two expressions involving relevant
velocities and equate to – 1.
Simplify: 3u2 75g2 10 3ug1u2 0, u2 100 3u75000
Answer
Marks
Guidance
4 4
M1
Simplify to quadratic in u (may see g ).
u5 3g
A1
OE. Accept 50 3 or 86.6.
5
Answer
Marks
Guidance
Question
Answer
Marks
Question 3:
3 | usin605g, and ucos60 , or usin6015g , and ucos60 | B1
usin605g
If is direction of velocity at t 5, tan
ucos60 | M1* | Accept equivalent for t15.
usin605g usin6015g
For perpendicular directions, 1
ucos60 ucos60 | M1dep | Multiply two expressions involving relevant
velocities and equate to – 1.
Simplify: 3u2 75g2 10 3ug1u2 0, u2 100 3u75000
4 4 | M1 | Simplify to quadratic in u (may see g ).
u5 3g | A1 | OE. Accept 50 3 or 86.6.
5
Question | Answer | Marks | Guidance
At time $t = 0$ seconds, a particle $P$ is projected with speed $u$ m s$^{-1}$ at an angle $60°$ above the horizontal from a point $O$. In the subsequent motion $P$ moves freely under gravity. The direction of motion of $P$ when $t = 5$ is perpendicular to its direction of motion when $t = 15$.
Find the value of $u$. [5]
\hfill \mbox{\textit{CAIE Further Paper 3 2024 Q3 [5]}}