Perpendicular velocity directions

Edexcel M2 2014 June Q6

13 marks Standard +0.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{82cadc37-4cb0-455e-9531-e09ec0c19533-11_711_917_219_561} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} A particle $P$ is projected from a point $A$ with speed $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation $\alpha$, where $\sin \alpha = \frac { 4 } { 5 }$. The point $A$ is 10 m vertically above the point $O$ which is on horizontal ground, as shown in Figure 4. The particle $P$ moves freely under gravity and reaches the ground at the point $B$. Calculate

the greatest height above the ground of $P$, as it moves from $A$ to $B$,
the distance $O B$. The point $C$ lies on the path of $P$. The direction of motion of $P$ at $C$ is perpendicular to the direction of motion of $P$ at $A$.
Find the time taken by $P$ to move from $A$ to $C$.

Edexcel M2 2018 June Q7

15 marks Challenging +1.2

7. A particle, of mass 0.3 kg , is projected from a point $O$ on horizontal ground with speed $u$. The particle is projected at an angle $\alpha$ above the horizontal, where $\tan \alpha = 2$, and moves freely under gravity. When the particle has moved a horizontal distance $x$ from $O$, its height above the ground is $y$.

Show that $$y = 2 x - \frac { 5 g } { 2 u ^ { 2 } } x ^ { 2 }$$ The particle hits the ground at the point $A$, where $O A = 36 \mathrm {~m}$.
Find $u$, the speed of projection.
Find the minimum kinetic energy of the particle as it moves between $O$ and $A$. The point $B$ lies on the path of the particle. The direction of motion of the particle at $B$ is perpendicular to the initial direction of motion of the particle.
Find the horizontal distance between $O$ and $B$.

CAIE Further Paper 3 2024 June Q3

5 marks Challenging +1.2

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]

CAIE Further Paper 3 2024 June Q6

8 marks Standard +0.8

A particle $P$ is projected with speed $u\text{ ms}^{-1}$ at an angle $\theta$ above the horizontal from a point $O$ and moves freely under gravity. After 5 seconds the speed of $P$ is $\frac{3}{4}u$.

Show that $\frac{7}{16}u^2 - 100u\sin\theta + 2500 = 0$. [3]
It is given that the velocity of $P$ after 5 seconds is perpendicular to the initial velocity. Find, in either order, the value of $u$ and the value of $\sin\theta$. [5]

CAIE Further Paper 3 2021 November Q5

7 marks Challenging +1.8

A particle $P$ is projected from a point $O$ on a horizontal plane and moves freely under gravity. Its initial speed is $u$ ms$^{-1}$ and its angle of projection is $\sin^{-1}(\frac{3}{5})$ above the horizontal. At time 8 s after projection, $P$ is at the point $A$. At time 32 s after projection, $P$ is at the point $B$. The direction of motion of $P$ at $B$ is perpendicular to its direction of motion at $A$. Find the value of $u$. [7]

CAIE Further Paper 3 2021 November Q1

5 marks Moderate -0.8

A particle is projected with speed $u$ at an angle $\alpha$ above the horizontal from a point $O$ on a horizontal plane. The particle moves freely under gravity.

Write down the horizontal and vertical components of the velocity of the particle at time $T$ after projection. [2] At time $T$ after projection, the direction of motion of the particle is perpendicular to the direction of projection.
Express $T$ in terms of $u$, $g$ and $\alpha$. [2]
Deduce that $T > \frac{u}{g}$. [1]