Energy methods in projectiles

7. \begin{figure}[h] \end{figure} At time \(t = 0\) a particle \(P\) is projected from a fixed point \(A\) on horizontal ground. The particle is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the ground. The particle moves freely under gravity. At time \(t = 3\) seconds, \(P\) is passing through the point \(B\) with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving downwards at an angle \(\beta\) to the horizontal, as shown in Figure 5.

1. By considering energy, find the height of \(B\) above the ground.
2. Find the size of angle \(\alpha\).
3. Find the size of angle \(\beta\).
4. Find the least speed of \(P\) as \(P\) travels from \(A\) to \(B\). As \(P\) travels from \(A\) to \(B\), the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is such that \(v \leqslant 15\) for an interval of \(T\) seconds.
5. Find the value of \(T\).
   \section*{\textbackslash section*\{Question 7 continued\}}

7. \begin{figure}[h] \end{figure} The fixed point \(A\) is 20 m vertically above the point \(O\) which is on horizontal ground. At time \(t = 0\), a particle \(P\) is projected from \(A\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) above the horizontal. The particle moves freely under gravity. At time \(t = 5\) seconds, \(P\) strikes the ground at the point \(B\), where \(O B = 40 \mathrm {~m}\), as shown in Figure 4.

1. By considering energy, find the speed of \(P\) as it hits the ground at \(B\).
2. Find the least speed of \(P\) as it moves from \(A\) to \(B\).
3. Find the length of time for which the speed of \(P\) is more than \(10 \mathrm {~ms} ^ { - 1 }\).

6. \begin{figure}[h] \end{figure} A small ball is projected with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) on horizontal ground. The angle of projection is \(\alpha\) above the horizontal. A horizontal platform is at height \(h\) metres above the ground. The ball moves freely under gravity until it hits the platform at the point B, as shown in Figure 2. The speed of the ball immediately before it hits the platform at \(B\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(h\). Given that \(\sin \alpha = 0.85\),
2. find the horizontal distance from \(A\) to \(B\).

A particle \(A\) of mass 0.5 kg is projected vertically upwards from horizontal ground with speed 25 m s\(^{-1}\).

1. Find the speed of \(A\) when it reaches a height of 20 m above the ground. [2]

When \(A\) reaches a height of 20 m, it collides with a particle \(B\) of mass 0.3 kg which is moving downwards in the same vertical line as \(A\) with speed 32.5 m s\(^{-1}\). In the collision between the two particles, \(B\) is brought to instantaneous rest.

1. Show that the velocity of \(A\) immediately after the collision is 4.5 m s\(^{-1}\) downwards. [2]
2. Find the time interval between \(A\) and \(B\) reaching the ground. You should assume that \(A\) does not bounce when it reaches the ground. [4]

[In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) being vertically upwards.] \includegraphics{figure_3} The point \(O\) is a fixed point on a horizontal plane. A ball is projected from \(O\) with velocity \((3\mathbf{i} + v\mathbf{j})\) m s\(^{-1}\) where \(v > 3\). The ball moves freely under gravity and passes through the point \(A\) before reaching its maximum height above the horizontal plane, as shown in Figure 3. The ball passes through \(A\) at time \(\frac{15}{49}\) s after projection. The initial kinetic energy of the ball is \(E\) joules. When the ball is at \(A\) it has kinetic energy \(\frac{1}{2}E\) joules.

1. Find the value of \(v\). [8]

At another point \(B\) on the path of the ball the kinetic energy is also \(\frac{1}{2}E\) joules. The ball passes through \(B\) at time \(T\) seconds after projection.

1. Find the value of \(T\). [3]

\includegraphics{figure_3} A particle \(P\) is projected from a point \(A\) with speed \(u\) m s\(^{-1}\) at an angle of elevation \(\theta\), where \(\cos \theta = \frac{4}{5}\). The point \(B\), on horizontal ground, is vertically below \(A\) and \(AB = 45\) m. After projection, \(P\) moves freely under gravity passing through a point \(C\), 30 m above the ground, before striking the ground at the point \(D\), as shown in Figure 3. Given that \(P\) passes through \(C\) with speed 24.5 m s\(^{-1}\),

1. using conservation of energy, or otherwise, show that \(u = 17.5\), [4]
2. find the size of the angle which the velocity of \(P\) makes with the horizontal as \(P\) passes through \(C\), [3]
3. find the distance \(BD\). [7]

\includegraphics{figure_1} The points \(O\) and \(B\) are on horizontal ground. The point \(A\) is \(h\) metres vertically above \(O\). A particle \(P\) is projected from \(A\) with speed 12 m s\(^{-1}\) at an angle \(\alpha°\) to the horizontal. The particle moves freely under gravity and hits the ground at \(B\), as shown in Figure 1. The speed of \(P\) immediately before it hits the ground is 15 m s\(^{-1}\).

1. By considering energy, find the value of \(h\). [4]

Given that 1.5 s after it is projected from \(A\), \(P\) is at a point 4 m above the level of \(A\), find

1. the value of \(\alpha\), [3]
2. the direction of motion of \(P\) immediately before it reaches \(B\). [3]

\includegraphics{figure_1} A smooth solid hemisphere, of radius 0.8 m and centre \(O\), is fixed with its plane face on a horizontal table. A particle of mass 0.5 kg is projected horizontally with speed \(u\) m s\(^{-1}\) from the highest point \(A\) of the hemisphere. The particle leaves the hemisphere at the point \(B\), which is a vertical distance of 0.2 m below the level of \(A\). The speed of the particle at \(B\) is \(v\) m s\(^{-1}\) and the angle between \(OA\) and \(OB\) is \(\theta\), as shown in Fig. 1.

1. Find the value of \(\cos \theta\). [1]
2. Show that \(v^2 = 5.88\). [3]
3. Find the value of \(u\). [3]