3.02i Projectile motion: constant acceleration model

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. \begin{figure}[h] \end{figure} A small ball \(P\) is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) which is 47.5 m above a horizontal beach. The ball moves freely under gravity and hits the beach at the point \(B\), as shown in Figure 3.

1. By considering energy, find the speed of \(P\) immediately before it hits the beach. The ball was projected from \(A\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\)
2. Find the greatest height above the beach of \(P\) as it moved from \(A\) to \(B\).
3. Find the least speed of \(P\) as it moved between \(A\) and \(B\).
4. Find the horizontal distance from \(A\) to \(B\).

4. At time \(t = 0\) a ball is projected from a fixed point \(A\) on horizontal ground to hit a target. The ball is projected from \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) to the horizontal. At time \(t = 2 \mathrm {~s}\) the ball hits the target. At the instant when it hits the target, the ball is travelling downwards at \(30 ^ { \circ }\) below the horizontal with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball is modelled as a particle moving freely under gravity and the target is modelled as the point \(T\).

1. Find
   1. the value of \(\theta\),
   2. the value of \(u\). The height of \(T\) above the ground is \(h\) metres.
2. Find the value of \(h\).
3. Find the length of time for which the ball is more than \(h\) metres above the ground during the flight from \(A\) to \(T\).

8. \begin{figure}[h] \end{figure} The fixed point \(A\) is \(h\) metres vertically above the point \(O\) that is on horizontal ground. At time \(t = 0\), a particle \(P\) is projected from \(A\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle moves freely under gravity. At time \(t = 2.5\) seconds, \(P\) strikes the ground at the point \(B\). At the instant when \(P\) strikes the ground, the speed of \(P\) is \(18 \mathrm {~ms} ^ { - 1 }\), as shown in Figure 4.

1. By considering energy, find the value of \(h\).
2. Find the distance \(O B\). As \(P\) moves from \(A\) to \(B\), the speed of \(P\) is less than or equal to \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T\) seconds.
3. Find the value of \(T\)

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} A rough ramp is fixed to horizontal ground.
The ramp is inclined to the ground at an angle \(\alpha\), where \(\tan \alpha = \frac { 7 } { 24 }\) The point \(A\) is at the bottom of the ramp and the point \(B\) is at the top of the ramp. The line \(A B\) is a line of greatest slope of the ramp and \(A B = 15 \mathrm {~m}\), as shown in Figure 3. A particle \(P\) of mass 0.3 kg is projected with speed \(U \mathrm {~ms} ^ { - 1 }\) from \(A\) directly towards \(B\). At the instant \(P\) reaches the point \(B\), the velocity of \(P\) is \(( 24 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The particle leaves the ramp at \(B\), and moves freely under gravity until it hits the horizontal ground at the point \(C\).
The coefficient of friction between \(P\) and the ramp is \(\frac { 1 } { 5 }\)

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
2. Use the work-energy principle to find the value of \(U\).
3. Find the time taken by \(P\) to move from \(B\) to \(C\). At the instant immediately before \(P\) hits the ground at \(C\), the particle is moving downwards at \(\theta ^ { \circ }\) to the horizontal.
4. Find the value of \(\theta\)

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 \(T _ { 2 }\)

7. \begin{figure}[h] \end{figure} At time \(t = 0\), a particle \(P\) of mass 0.7 kg is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) at an angle \(\theta ^ { \circ }\) to the horizontal. The particle moves freely under gravity. At time \(t = 2\) seconds, \(P\) passes through the point \(A\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving downwards at \(45 ^ { \circ }\) to the horizontal, as shown in Figure 4. Find

1. the value of \(\theta\),
2. the kinetic energy of \(P\) as it reaches the highest point of its path. For an interval of \(T\) seconds, the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is such that \(v \leqslant 6\)
3. Find the value of \(T\).

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7. \begin{figure}[h] \end{figure} A ball is projected with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(P\) on a cliff above horizontal ground. The point O on the ground is vertically below P and OP is 36 m . The ball is projected at an angle \(\theta ^ { \circ }\) to the horizontal. The point Q is the highest point of the path of the ball and is 12 m above the level of P. The ball moves freely under gravity and hits the ground at the point R , as shown in Figure 3. Find

1. the value of \(\theta\),
2. the distance OR ,
3. the speed of the ball as it hits the ground at R.

5. A particle \(P\) is projected with velocity \(( 2 u \mathbf { i } + 3 u \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) from a point \(O\) on a horizontal plane, where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical unit vectors respectively. The particle \(P\) strikes the plane at the point \(A\) which is 735 m from \(O\).

1. Show that \(u = 24.5\).
2. Find the time of flight from \(O\) to \(A\). The particle \(P\) passes through a point \(B\) with speed \(65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the height of \(B\) above the horizontal plane.

7. \begin{figure}[h] \end{figure} A particle \(P\) is projected from a point \(A\) with speed \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\). The point \(O\) is on horizontal ground, with \(O\) vertically below \(A\) and \(O A = 20 \mathrm {~m}\). The particle \(P\) moves freely under gravity and passes through a point \(B\), which is 16 m above ground, before reaching the ground at the point \(C\), as shown in Figure 4. Calculate

1. the time of the flight from \(A\) to \(C\),
2. the distance \(O C\),
3. the speed of \(P\) at \(B\),
4. the angle that the velocity of \(P\) at \(B\) makes with the horizontal.

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. \begin{figure}[h] \end{figure} A ball is thrown from a point \(O\), which is 6 m above horizontal ground. The ball is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. There is a thin vertical post which is 4 m high and 8 m horizontally away from the vertical through \(O\), as shown in Figure 2. The ball passes just above the top of the post 2 s after projection. The ball is modelled as a particle.

1. Show that \(\tan \theta = 2.2\)
2. Find the value of \(u\). The ball hits the ground \(T\) seconds after projection.
3. Find the value of \(T\). Immediately before the ball hits the ground the direction of motion of the ball makes an angle \(\alpha\) with the horizontal.
4. Find \(\alpha\).

7. \begin{figure}[h] \end{figure} In a ski-jump competition, a skier of mass 80 kg moves from rest at a point \(A\) on a ski-slope. The skier's path is an arc \(A B\). The starting point \(A\) of the slope is 32.5 m above horizontal ground. The end \(B\) of the slope is 8.1 m above the ground. When the skier reaches \(B\), she is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and moving upwards at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Fig. 2. The distance along the slope from \(A\) to \(B\) is 60 m . The resistance to motion while she is on the slope is modelled as a force of constant magnitude \(R\) newtons. By using the work-energy principle,

1. find the value of \(R\). On reaching \(B\), the skier then moves through the air and reaches the ground at the point \(C\). The motion of the skier in moving from \(B\) to \(C\) is modelled as that of a particle moving freely under gravity.
2. Find the time for the skier to move from \(B\) to \(C\).
3. Find the horizontal distance from \(B\) to \(C\).
4. Find the speed of the skier immediately before she reaches \(C\). END

6. \begin{figure}[h] \end{figure} A golf ball \(P\) is projected with speed \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) on a cliff above horizontal ground. The angle of projection is \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 4 } { 3 }\). The ball moves freely under gravity and hits the ground at the point \(B\), as shown in Figure 4.

1. Find the greatest height of \(P\) above the level of \(A\). The horizontal distance from \(A\) to \(B\) is 168 m .
2. Find the height of \(A\) above the ground. By considering energy, or otherwise,
3. find the speed of \(P\) as it hits the ground at \(B\).

7. \begin{figure}[h] \end{figure} A ball is thrown from a point \(A\) at a target, which is on horizontal ground. The point \(A\) is 12 m above the point \(O\) on the ground. The ball is thrown from \(A\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) below the horizontal. The ball is modelled as a particle and the target as a point \(T\). The distance \(O T\) is 15 m . The ball misses the target and hits the ground at the point \(B\), where \(O T B\) is a straight line, as shown in Figure 4. Find

1. the time taken by the ball to travel from \(A\) to \(B\),
2. the distance \(T B\). The point \(X\) is on the path of the ball vertically above \(T\).
3. Find the speed of the ball at \(X\).

6. \begin{figure}[h] \end{figure} A child playing cricket on horizontal ground hits the ball towards a fence 10 m away. The ball moves in a vertical plane which is perpendicular to the fence. The ball just passes over the top of the fence, which is 2 m above the ground, as shown in Figure 3. The ball is modelled as a particle projected with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from point \(O\) on the ground at an angle \(\alpha\) to the ground.

1. By writing down expressions for the horizontal and vertical distances, from \(O\) of the ball \(t\) seconds after it was hit, show that $$2 = 10 \tan \alpha - \frac { 50 g } { u ^ { 2 } \cos ^ { 2 } \alpha }$$ Given that \(\alpha = 45 ^ { \circ }\),
2. find the speed of the ball as it passes over the fence.

7. \begin{figure}[h] \end{figure} A small stone is projected from a point \(O\) at the top of a vertical cliff \(O A\). The point \(O\) is 52.5 m above the sea. The stone rises to a maximum height of 10 m above the level of \(O\) before hitting the sea at the point \(B\), where \(A B = 50 \mathrm {~m}\), as shown in Figure 4. The stone is modelled as a particle moving freely under gravity.

1. Show that the vertical component of the velocity of projection of the stone is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the speed of projection.
3. Find the time after projection when the stone is moving parallel to \(O B\).

6. \begin{figure}[h] \end{figure} A ball is projected from a point \(A\) which is 8 m above horizontal ground as shown in Figure 4. The ball is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) above the horizontal. The ball moves freely under gravity and hits the ground at the point \(B\). The speed of the ball immediately before it hits the ground is \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy, find the value of \(u\). The time taken for the ball to move from \(A\) to \(B\) is 2 seconds. Find
2. the value of \(\theta\),
3. the minimum speed of the ball on its path from \(A\) to \(B\).

6. \begin{figure}[h] \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

1. the greatest height above the ground of \(P\), as it moves from \(A\) to \(B\),
2. 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 \(A\).
3. Find the time taken by \(P\) to move from \(A\) to \(C\).

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 \(B\).

7. \begin{figure}[h] \end{figure} At time \(t = 0\), a particle is projected from a fixed point \(O\) on horizontal ground with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) to the horizontal. The particle moves freely under gravity and passes through the point \(A\) when \(t = 4 \mathrm {~s}\). As it passes through \(A\), the particle is moving upwards at \(20 ^ { \circ }\) to the horizontal with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Figure 3.

1. Find the value of \(u\) and the value of \(\theta\). At the point \(B\) on its path the particle is moving downwards at \(20 ^ { \circ }\) to the horizontal with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the time taken for the particle to move from \(A\) to \(B\). The particle reaches the ground at the point \(C\).
3. Find the distance \(O C\).

6. \begin{figure}[h] \end{figure} The points \(A\) and \(B\) lie 40 m apart on horizontal ground. At time \(t = 0\) the particles \(P\) and \(Q\) are projected in the vertical plane containing \(A B\) and move freely under gravity. Particle \(P\) is projected from \(A\) with speed \(30 \mathrm {~ms} ^ { - 1 }\) at \(60 ^ { \circ }\) to \(A B\) and particle \(Q\) is projected from \(B\) with speed \(q \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at angle \(\theta\) to \(B A\), as shown in Figure 4. At \(t = 2\) seconds, \(P\) and \(Q\) collide.

1. Find
   1. the size of angle \(\theta\),
   2. the value of \(q\).
2. Find the speed of \(P\) at the instant before it collides with \(Q\).

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 \(y\).

1. Show that $$y = 2 x - \frac { 5 g } { 2 u ^ { 2 } } x ^ { 2 }$$ The particle hits the ground at the point \(A\), where \(O A = 36 \mathrm {~m}\).
2. Find \(u\), the speed of projection.
3. 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.
4. Find the horizontal distance between \(O\) and \(B\).