Vector form projectile motion

4 A projectile P travels in a vertical plane over level ground. Its position vector \(\mathbf { r }\) at time \(t\) seconds after projection is modelled by $$\mathbf { r } = \binom { x } { y } = \binom { 0 } { 5 } + \binom { 30 } { 40 } t - \binom { 0 } { 5 } t ^ { 2 } ,$$ where distances are in metres and the origin is a point on the level ground.

1. Write down
   (A) the height from which P is projected,
   (B) the value of \(g\) in this model.
2. Find the displacement of P from \(t = 3\) to \(t = 5\).
3. Show that the equation of the trajectory is $$y = 5 + \frac { 4 } { 3 } x - \frac { x ^ { 2 } } { 180 } .$$

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors, with \(\mathbf { i }\) horizontal and \(\mathbf { j }\) vertical.]

\begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) lie on horizontal ground.
At time \(t = 0\), a particle \(P\) is projected from \(A\) with velocity \(( 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Particle \(P\) moves freely under gravity and hits the ground at \(B\), as shown in Figure 6 .

1. Find the distance \(A B\). The speed of \(P\) is less than \(5 \mathrm {~ms} ^ { - 1 }\) for an interval of length \(T\) seconds.
2. Find the value of \(T\) At the instant when the direction of motion of \(P\) is perpendicular to the initial direction of motion of \(P\), the particle is \(h\) metres above the ground.
3. Find the value of \(h\).

6. [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.] At \(t = 0\) a particle \(P\) is projected from a fixed point \(O\) with velocity ( \(7 \mathbf { i } + 7 \sqrt { 3 } \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity. The position vector of a point on the path of \(P\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { m }\) relative to \(O\).

1. Show that $$y = \sqrt { 3 } x - \frac { g } { 98 } x ^ { 2 }$$
2. Find the direction of motion of \(P\) when it passes through the point on the path where \(x = 20\) At time \(T\) seconds \(P\) passes through the point with position vector \(( 2 \lambda \mathbf { i } + \lambda \mathbf { j } ) \mathrm { m }\) where \(\lambda\) is a positive constant.
3. Find the value of \(T\).
   DO NOT WIRITE IN THIS AREA

8. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, with \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] \begin{figure}[h] \end{figure} At time \(t = 0\), a small ball is projected from a fixed point \(O\) on horizontal ground. The ball is projected from \(O\) with velocity ( \(p \mathbf { i } + q \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\), where \(p\) and \(q\) are positive constants. The ball moves freely under gravity. At time \(t = 3\) seconds, the ball passes through the point \(A\) with velocity ( \(8 \mathbf { i } - 12 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\), as shown in Figure 4.

1. Find the speed of the ball at the instant it is projected from \(O\). For an interval of \(T\) seconds the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the ball is such that \(v \leqslant 10\)
2. Find the value of \(T\). At the point \(B\) on the path of the ball, the direction of motion of the ball is perpendicular to the direction of motion of the ball at \(A\).
3. Find the vertical height of \(B\) above \(A\).

1. \hspace{0pt} [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.]

\begin{figure}[h] \end{figure} A small ball is projected with velocity \(( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) from a fixed point \(A\) on horizontal ground. The ball hits the ground at the point \(B\), as shown in Figure 5. The motion of the ball is modelled as a particle moving freely under gravity.

1. Find the distance \(A B\). When the height of the ball above the ground is more than \(h\) metres, the speed of the ball is less than \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find the smallest possible value of \(h\). When the ball is at the point \(C\) on its path, the direction of motion of the ball is perpendicular to the direction of motion of the ball at the instant before it hits the ground at \(B\).
3. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of the ball when it is at \(C\).

1. \hspace{0pt} [In this question, the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane with \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.]

\begin{figure}[h] \end{figure} A small ball is projected with velocity \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) from the fixed point \(A\).
The point \(A\) is 20 m above horizontal ground.
The ball hits the ground at the point \(B\), as shown in Figure 4.
The ball is modelled as a particle moving freely under gravity.

1. By considering energy, find the speed of the ball at the instant immediately before it hits the ground.
2. Find the direction of motion of the ball at the instant immediately before it hits the ground.
3. Find the time taken for the ball to travel from \(A\) to \(B\). At the instant when the direction of motion of the ball is perpendicular to ( \(3 \mathbf { i } + 2 \mathbf { j }\) ) the ball is \(h\) metres above the ground.
4. Find the value of \(h\).

1. \hspace{0pt} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and j being vertically upwards.]

\begin{figure}[h] \end{figure} A golf ball is hit from a point \(O\) on horizontal ground and is modelled as a particle moving freely under gravity. The initial velocity of the ball is \(( 2 u \mathbf { i } + u \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The ball first hits the horizontal ground at a point which is 80 m from \(O\), as shown in Figure 3. Use the model to

1. show that \(u = 14\)
2. find the total time, while the ball is in the air, for which the speed of the ball is greater than \(7 \sqrt { 17 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

7. [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. Position vectors are given relative to a fixed origin O.] At time \(t = 0\) seconds, the particle \(P\) is projected from \(O\) with velocity ( \(3 \mathbf { i } + \lambda \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\), where \(\lambda\) is a positive constant. The particle moves freely under gravity. As \(P\) passes through the fixed point \(A\) it has velocity \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The kinetic energy of \(P\) at the instant it passes through \(A\) is half the initial kinetic energy of \(P\). Find the position vector of \(A\), giving the components to 2 significant figures.
(10)

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors, with \(\mathbf { i }\) horizontal and \(\mathbf { j }\) vertical.]

\begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) lie on horizontal ground.
At time \(t = 0\), a particle \(P\) is projected from \(A\) with velocity ( \(4 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) Particle \(P\) moves freely under gravity and hits the ground at \(B\), as shown in Figure 3 .
At time \(T _ { 1 }\) seconds, \(P\) is at its highest point above the ground.

1. Find the value of \(T _ { 1 }\) At time \(t = 0\), a particle \(Q\) is also projected from \(A\) but with velocity \(( 5 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Particle \(Q\) moves freely under gravity.
2. Find the vertical distance between \(Q\) and \(P\) at time \(T _ { 1 }\) seconds, giving your answer to 2 significant figures. At the instant when particle \(P\) reaches \(B\), particle \(Q\) is moving at \(\alpha ^ { \circ }\) below the horizontal.
3. Find the value of \(\alpha\). At time \(T _ { 2 }\) seconds, the direction of motion of \(Q\) is perpendicular to the initial direction of motion of \(Q\).
4. Find the value of \(T _ { 2 }\)

6. \begin{figure}[h] \end{figure} A cricket ball is hit from a point \(A\) with velocity of \(( p \mathbf { i } + q \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), at an angle \(\alpha\) above the horizontal. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are respectively horizontal and vertically upwards. The point \(A\) is 0.9 m vertically above the point \(O\), which is on horizontal ground. The ball takes 3 seconds to travel from \(A\) to \(B\), where \(B\) is on the ground and \(O B = 57.6 \mathrm {~m}\), as shown in Figure 3. By modelling the motion of the cricket ball as that of a particle moving freely under gravity,

1. find the value of \(p\),
2. show that \(q = 14.4\),
3. find the initial speed of the cricket ball,
4. find the exact value of \(\tan \alpha\).
5. Find the length of time for which the cricket ball is at least 4 m above the ground.
6. State an additional physical factor which may be taken into account in a refinement of the above model to make it more realistic.

1. \hspace{0pt} [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.]

\begin{figure}[h] \end{figure} At time \(t = 0\), a particle \(P\) is projected from the point \(A\) which has position vector 10j metres with respect to a fixed origin \(O\) at ground level. The ground is horizontal. The velocity of projection of \(P\) is \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), as shown in Figure 3. The particle moves freely under gravity and reaches the ground after \(T\) seconds.

1. For \(0 \leqslant t \leqslant T\), show that, with respect to \(O\), the position vector, \(\mathbf { r }\) metres, of \(P\) at time \(t\) seconds is given by $$\mathbf { r } = 3 t \mathbf { i } + \left( 10 + 5 t - 4.9 t ^ { 2 } \right) \mathbf { j }$$
2. Find the value of \(T\).
3. Find the velocity of \(P\) at time \(t\) seconds \(( 0 \leqslant t \leqslant T )\). When \(P\) is at the point \(B\), the direction of motion of \(P\) is \(45 ^ { \circ }\) below the horizontal.
4. Find the time taken for \(P\) to move from \(A\) to \(B\).
5. Find the speed of \(P\) as it passes through \(B\).

6. [In this question, \(\mathbf { i }\) is a horizontal unit vector and \(\mathbf { j }\) is an upward vertical unit vector.] A particle \(P\) is projected from a fixed origin \(O\) with velocity ( \(3 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity and passes through the point \(A\) with position vector \(\lambda ( \mathbf { i } - \mathbf { j } ) \mathrm { m }\), where \(\lambda\) is a positive constant.

1. Find the value of \(\lambda\).
2. Find
   1. the speed of \(P\) at the instant when it passes through \(A\),
   2. the direction of motion of \(P\) at the instant when it passes through \(A\).
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      VILV SIHI NI JAHM ION OC
      VJ4V SIHI NI JIIYM ION OC

7 In this question the \(x\) - and \(y\)-directions are horizontal and vertically upwards respectively and the origin is on horizontal ground.
A ball is thrown from a point 5 m above the origin with an initial velocity \(\binom { 14 } { 7 } \mathrm {~ms} ^ { - 1 }\).

1. Find the position vector of the ball at time \(t \mathrm {~s}\) after it is thrown.
2. Find the distance between the origin and the point at which the ball lands on the ground.

6 A ball is hit from horizontal ground with velocity \(( 10 \mathbf { i } + 24.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) where the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertically upwards respectively.

1. State two assumptions that you should make about the ball in order to make predictions about its motion.
2. The path of the ball is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-5_351_771_705_625}
   1. Show that the time of flight of the ball is 5 seconds.
   2. Find the range of the ball.
3. In fact the ball hits a vertical wall that is 20 metres from the initial position of the ball. \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-5_351_403_1466_769} Find the height of the ball when it hits the wall.
4. If a heavier ball were projected in the same way, would your answers to part (b) of this question change? Explain why.

7 A projectile P travels in a vertical plane over level ground. Its position vector \(\mathbf { r }\) at time \(t\) seconds after projection is modelled by $$\mathbf { r } = \binom { x } { y } = \binom { 0 } { 5 } + \binom { 30 } { 40 } t - \binom { 0 } { 5 } t ^ { 2 }$$ where distances are in metres and the origin is a point on the level ground.

1. Write down
   (A) the height from which P is projected,
   (B) the value of \(g\) in this model.
2. Find the displacement of P from \(t = 3\) to \(t = 5\).
3. Show that the equation of the trajectory is $$y = 5 + \frac { 4 } { 3 } x - \frac { x ^ { 2 } } { 180 }$$

8. A particle \(P\) is projected from a point \(O\) with initial velocity \(( 3 \cdot 5 \mathbf { i } + 12 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and moves under gravity. \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the horizontal and vertical directions respectively.

1. Find the initial speed of \(P\).
2. Show that the position vector \(\mathbf { r } \mathbf { m }\) of \(P\) at time \(t\) seconds after projection is given by $$\mathbf { r } = 3 \cdot 5 t \mathbf { i } + \left( 12 t - 4 \cdot 9 t ^ { 2 } \right) \mathbf { j } .$$
3. Find the horizontal distance of \(P\) from \(O\) at each of the times when it is 4.4 m vertically above the level of \(O\). In a refined model of the motion of \(P\), the position vector of \(P\) at time \(t\) seconds is taken to be $$\mathbf { r } = 3 \cdot 5 t \mathbf { i } + \left( 12 t - t ^ { 3 } \right) \mathbf { j } \mathbf { ~ m } .$$
4. Using this model, find the position vector of the highest point reached by \(P\).

6. A ball is projected with velocity \(( 4 w \mathbf { i } + 7 w \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) from the top of a vertical tower. After 5 seconds, the ball hits the ground at a point that is 60 m horizontally from the foot of the tower. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively.

1. Find the value of \(w\) and hence determine the height of the tower.
2. Determine the proportion of the 5 seconds for which the ball is on its way down.

[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.] At time \(t = 0\), a particle \(P\) is projected with velocity \((4\mathbf{i} + 9\mathbf{j})\) m s\(^{-1}\) from a fixed point \(O\) on horizontal ground. The particle moves freely under gravity. When \(P\) is at the point \(H\) on its path, \(P\) is at its greatest height above the ground.

1. Find the time taken by \(P\) to reach \(H\). [2]

At the point \(A\) on its path, the position vector of \(P\) relative to \(O\) is \((k\mathbf{i} + k\mathbf{j})\) m, where \(k\) is a positive constant.

1. Find the value of \(k\). [4]
2. Find, in terms of \(k\), the position vector of the other point on the path of \(P\) which is at the same vertical height above the ground as the point \(A\). [3]

At time \(T\) seconds the particle is at the point \(B\) and is moving perpendicular to \((4\mathbf{i} + 9\mathbf{j})\)

1. Find the value of \(T\). [4]

The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) lie in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) vertical. A ball of mass \(0.1\) kg is hit by a bat which gives it an impulse of \((3.5\mathbf{i} + 3\mathbf{j})\) Ns. The velocity of the ball immediately after being hit is \((10\mathbf{i} + 25\mathbf{j})\) m s\(^{-1}\).

1. Find the velocity of the ball immediately before it is hit. [3]

In the subsequent motion the ball is modelled as a particle moving freely under gravity. When it is hit the ball is 1 m above horizontal ground.

1. Find the greatest height of the ball above the ground in the subsequent motion. [3]

The ball is caught when it is again 1 m above the ground.

1. Find the distance from the point where the ball is hit to the point where it is caught. [4]

\includegraphics{figure_3} [In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) being vertical.] A particle \(P\) is projected from the point \(A\) which has position vector \(47.5\mathbf{j}\) metres with respect to a fixed origin \(O\). The velocity of projection of \(P\) is \((2u\mathbf{i} + 5u\mathbf{j})\) m s\(^{-1}\). The particle moves freely under gravity passing through the point \(B\) with position vector \(30\mathbf{i}\) metres, as shown in Figure 3.

1. Show that the time taken for \(P\) to move from \(A\) to \(B\) is 5 s. [6]
2. Find the value of \(u\). [2]
3. Find the speed of \(P\) at \(B\). [5]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in a horizontal and upward vertical direction respectively] A particle \(P\) is projected from a fixed point \(O\) on horizontal ground with velocity \(u(\mathbf{i} + c\mathbf{j}) \text{ ms}^{-1}\), where \(c\) and \(u\) are positive constants. The particle moves freely under gravity until it strikes the ground at \(A\), where it immediately comes to rest. Relative to \(O\), the position vector of a point on the path of \(P\) is \((x\mathbf{i} + y\mathbf{j})\) m.

1. Show that $$y = cx - \frac{4.9x^2}{u^2}.$$ [5]

Given that \(u = 7\), \(OA = R\) m and the maximum vertical height of \(P\) above the ground is \(H\) m,

1. using the result in part (a), or otherwise, find, in terms of \(c\),
   1. \(R\)
   2. \(H\).
   [6]

Given also that when \(P\) is at the point \(Q\), the velocity of \(P\) is at right angles to its initial velocity,

1. find, in terms of \(c\), the value of \(x\) at \(Q\). [6]

[In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively.] \includegraphics{figure_3} The point \(O\) is a fixed point on a horizontal plane. A ball is projected from \(O\) with velocity \((6\mathbf{i} + 12\mathbf{j})\) m s\(^{-1}\), and passes through the point \(A\) at time \(t\) seconds after projection. The point \(B\) is on the horizontal plane vertically below \(A\), as shown in Figure 3. It is given that \(OB = 2AB\). Find

1. the value of \(t\), [7]
2. the speed, \(V\) m s\(^{-1}\), of the ball at the instant when it passes through \(A\). [5]

At another point \(C\) on the path the speed of the ball is also \(V\) m s\(^{-1}\).

1. Find the time taken for the ball to travel from \(O\) to \(C\). [3]

At time \(t = 0\), a parachutist jumps out of an airplane that is travelling horizontally. The velocity, \(\mathbf{v}\) m s\(^{-1}\), of the parachutist at time \(t\) seconds is given by: $$\mathbf{v} = (40e^{-0.2t})\mathbf{i} + 50(e^{-0.2t} - 1)\mathbf{j}$$ The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. Assume that the parachutist is at the origin when \(t = 0\) Model the parachutist as a particle.

1. Find an expression for the position vector of the parachutist at time \(t\). [4 marks]
2. The parachutist opens her parachute when she has travelled 100 metres horizontally. Find the vertical displacement of the parachutist from the origin when she opens her parachute. [4 marks]
3. Carefully, explaining the steps that you take, deduce the value of \(g\) used in the formulation of this model. [3 marks]

In this question use \(g = 9.81\) m s\(^{-2}\). A ball is projected from the origin. After 2.5 seconds, the ball lands at the point with position vector \((40\mathbf{i} - 10\mathbf{j})\) metres. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. Assume that there are no resistance forces acting on the ball.

1. Find the speed of the ball when it is at a height of 3 metres above its initial position. [6 marks]
2. State the speed of the ball when it is at its maximum height. [1 mark]
3. Explain why the answer you found in part (b) may not be the actual speed of the ball when it is at its maximum height. [1 mark]