3.02h Motion under gravity: vector form

4 A ball \(B\) is projected from a point \(O\) on horizontal ground at an angle of \(40 ^ { \circ }\) above the horizontal. \(B\) hits the ground 1.8 s after the instant of projection. Calculate

1. the speed of projection of \(B\),
2. the greatest height of \(B\),
3. the distance from \(O\) of the point at which \(B\) hits the ground.

3 A particle \(P\) is projected with speed \(25 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. After 2 s the speed of \(P\) is \(15 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of \(\sin \theta\).
2. Find the range of the flight.

1 A particle is projected with speed \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal from a point on horizontal ground. Calculate the speed of the particle 2 s after the instant of projection.

3 A particle \(P\) is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upward displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and hence show that the equation of the path of \(P\) is \(y = x - 0.016 x ^ { 2 }\).
2. Calculate the horizontal distance between the two positions at which \(P\) is 2.4 m above the ground.

5 A particle \(P\) is projected with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. For the instant when the speed of \(P\) is \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and increasing,

1. show that the vertical component of the velocity of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards,
2. calculate the distance of \(P\) from \(O\).

3 \includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-2_397_1303_1790_422} The point \(O\) is 1.2 m below rough horizontal ground \(A B C\). A ball is projected from \(O\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(70 ^ { \circ }\) to the horizontal. The ball passes over the point \(A\) after travelling a horizontal distance of 2 m . The ball subsequently bounces once on the ground at \(B\). The ball leaves \(B\) with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and travels a further horizontal distance of 20 m before landing at \(C\) (see diagram).

1. Calculate the height above the level of \(O\) of the ball when it is vertically above \(A\).
2. Calculate the time after the instant of projection when the ball reaches \(B\).
3. Find the angle which the trajectory of the ball makes with the horizontal immediately after it bounces at \(B\). \includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-3_663_695_258_726} A cylinder of height 0.9 m and radius 0.9 m is placed symmetrically on top of a cylinder of height \(h \mathrm {~m}\) and radius \(r \mathrm {~m}\), where \(r < 0.9\), with plane faces in contact and axes in the same vertical line \(A B\), where \(A\) and \(B\) are centres of plane faces of the cylinders (see diagram). Both cylinders are uniform and made of the same material. The lower cylinder is gradually tilted and when the axis of symmetry is inclined at \(45 ^ { \circ }\) to the horizontal the upper cylinder is on the point of toppling without sliding.

4 A small ball \(B\) is projected from a point \(O\) with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal.

1. Calculate the speed and direction of motion of \(B\) for the instant 1.8 s after projection. The point \(O\) is 2 m above a horizontal plane.
2. Calculate the time after projection when \(B\) reaches the plane.

7 A particle \(P\) is projected with speed \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on a horizontal plane. In the subsequent motion, the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is $$y = k x - \frac { \left( 1 + k ^ { 2 } \right) x ^ { 2 } } { 245 }$$ where \(k\) is a constant. \(P\) passes through the points \(A ( 14 , a )\) and \(B ( 42,2 a )\), where \(a\) is a constant.

1. Calculate the two possible values of \(k\) and hence show that the larger of the two possible angles of projection is \(63.435 ^ { \circ }\), correct to 3 decimal places. For the larger angle of projection, calculate
2. the time after projection when \(P\) passes through \(A\),
3. the speed and direction of motion of \(P\) when it passes through \(B\). {www.cie.org.uk} after the live examination series. }

1 A stone \(S\) is thrown horizontally from the top \(T\) of a high tower. At the instant 1.6 s after \(S\) is thrown, the line \(S T\) makes an angle of \(30 ^ { \circ }\) below the horizontal. Find the speed with which \(S\) is thrown. [3]

4 A particle \(P\) is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. \(P\) subsequently bounces when it first strikes the ground at the point \(A\).

1. Find the time after projection when \(P\) first strikes the ground, and the distance \(O A\). When \(P\) bounces at \(A\) the horizontal component of the velocity of \(P\) is unchanged. The vertical component of velocity is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) immediately after bouncing. \(P\) strikes the ground for the second time at \(B\) where it remains at rest.
2. Calculate the first and last times after projection at which the speed of \(P\) is \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 A small ball is projected from a point 1.5 m above horizontal ground. At a point 9 m above the ground the ball is travelling at \(45 ^ { \circ }\) above the horizontal and its velocity is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the angle of projection of the ball.

4 A particle \(P\) is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and hence show that the equation of the trajectory of \(P\) is $$y = \frac { x } { \sqrt { 3 } } - \frac { 4 x ^ { 2 } } { 375 }$$
2. Find the horizontal distance between the two points at which \(P\) is 5 m above the ground.

7 A small ball \(B\) is projected from a point \(O\) which is \(h \mathrm {~m}\) above a horizontal plane. At time 2 s after projection \(B\) has speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving in the direction \(30 ^ { \circ }\) above the horizontal.

1. Find the initial speed and the angle of projection of \(B\). \(B\) has speed \(38 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) immediately before it strikes the plane.
2. Calculate \(h\). \(B\) bounces when it strikes the plane, and leaves the plane with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) but with its horizontal component of velocity unchanged.
3. Find the total time which elapses between the initial projection of \(B\) and the instant when it strikes the plane for the second time.

1 A small ball \(B\) is projected with speed \(38 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the horizontal from a point on horizontal ground. Find the speed of \(B\) when the path of \(B\) makes an angle of \(20 ^ { \circ }\) above the horizontal.

4 A small object is projected horizontally with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) above horizontal ground. At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and hence show that the equation of the path of the object is \(y = - \frac { 5 x ^ { 2 } } { V ^ { 2 } }\).
   The object passes through points with coordinates \(( a , - a )\) and \(\left( a ^ { 2 } , - 16 a \right)\), where \(a\) is a positive constant.
2. Find the value of \(a\).
3. Given that the object strikes the ground at the point where \(x = 5 a\), find the height of \(O\) above the ground .

2 A particle is projected from a point on horizontal ground with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. The particle strikes the ground 2 s after projection.

1. Find \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{4cd525d5-d59b-4ab9-85a3-fc3d97fd09fc-03_67_1571_438_328}
2. Calculate the time after projection at which the direction of motion of the particle is \(20 ^ { \circ }\) below the horizontal.

4 A small ball \(B\) is projected with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal from a point \(O\). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(B\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of the ball.
2. Find the value of \(x\) for which \(O B\) makes an angle of \(45 ^ { \circ }\) above the horizontal.

2 A small ball is projected from a point \(O\) on horizontal ground at an angle of \(30 ^ { \circ }\) above the horizontal. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of the ball from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. It is given that \(x = 40 t\).

1. Calculate the initial speed of the ball, and express \(y\) in terms of \(t\).
2. Hence find the equation of the trajectory of the ball.

2. A ball is thrown vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(P\) at height \(h\) metres above the ground. The ball hits the ground 0.75 s later. The speed of the ball immediately before it hits the ground is \(6.45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball is modelled as a particle.

1. Show that \(u = 0.9\)
2. Find the height above \(P\) to which the ball rises before it starts to fall towards the ground again.
3. Find the value of \(h\).

1. A stone is projected vertically upwards from a point \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After projection the stone moves freely under gravity until it returns to \(A\). The time between the instant that the stone is projected and the instant that it returns to \(A\) is \(3 \frac { 4 } { 7 }\) seconds.

Modelling the stone as a particle,

1. show that \(u = 17 \frac { 1 } { 2 }\),
2. find the greatest height above \(A\) reached by the stone,
3. find the length of time for which the stone is at least \(6 \frac { 3 } { 5 } \mathrm {~m}\) above \(A\).

3. A competitor makes a dive from a high springboard into a diving pool. She leaves the springboard vertically with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards. When she leaves the springboard, she is 5 m above the surface of the pool. The diver is modelled as a particle moving vertically under gravity alone and it is assumed that she does not hit the springboard as she descends. Find

1. her speed when she reaches the surface of the pool,
2. the time taken to reach the surface of the pool.
3. State two physical factors which have been ignored in the model.

2. At time \(t = 0\), a particle is projected vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 10 m above the ground. At time \(T\) seconds, the particle hits the ground with speed \(17.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the value of \(u\),
2. the value of \(T\).