3.02h Motion under gravity: vector form

\includegraphics{figure_7} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a smooth plane. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). \(P\) lies on the plane and \(Q\) hangs vertically below the pulley at a height of 0.8 m above the floor (see diagram). The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. \(P\) is released from rest and \(Q\) moves vertically downwards.

1. Find the tension in the string and the magnitude of the acceleration of the particles. [5]

\(Q\) hits the floor and does not bounce. It is given that \(P\) does not reach the pulley in the subsequent motion.

1. Find the time, from the instant at which \(P\) is released, for \(Q\) to reach the floor. [2]
2. When \(Q\) hits the floor the string becomes slack. Find the time, from the instant at which \(P\) is released, for the string to become taut again. [4]

A particle of mass 0.5 kg starts from rest and slides down a line of greatest slope of a smooth plane. The plane is inclined at an angle of 30° to the horizontal.

1. Find the time taken for the particle to reach a speed of 2.5 m s\(^{-1}\). [3]
2. Find the distance that the particle travels along the ground before it comes to rest. [3]

When the particle has travelled 3 m down the slope from its starting point, it reaches rough horizontal ground at the bottom of the slope. The frictional force acting on the particle is 1 N.

\includegraphics{figure_7} A small object is projected with speed \(24\text{ m s}^{-1}\) from a point \(O\) at the foot of a plane inclined at \(45°\) to the horizontal. The angle of projection of the object is \(15°\) above a line of greatest slope of the plane (see diagram). At time \(t\text{ s}\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x\text{ m}\) and \(y\text{ m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\), and hence find the value of \(t\) for the instant when the object strikes the plane. [4]
2. Express the vertical height of the object above the plane in terms of \(t\) and hence find the greatest vertical height of the object above the plane. [5]

A smooth sphere with centre \(O\) and of radius \(a\) is fixed to a horizontal plane. A particle \(P\) of mass \(m\) is projected horizontally from the highest point of the sphere with speed \(u\), so that it begins to move along the surface of the sphere. The particle \(P\) loses contact with the sphere at the point \(Q\) on the sphere, where \(OQ\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\cos\theta = \frac{u^2 + 2ag}{3ag}\). [4]

It is given that \(\cos\theta = \frac{5}{9}\).

1. Find, in terms of \(a\) and \(g\), an expression for the vertical component of the velocity of \(P\) just before it hits the horizontal plane to which the sphere is fixed. [3]
2. Find an expression for the time taken by \(P\) to fall from \(Q\) to the plane. Give your answer in the form \(k\sqrt{\frac{a}{g}}\), stating the value of \(k\) correct to 3 significant figures. [2]

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 \(\frac{3}{4}u\).

1. Show that \(\frac{7}{16}u^2 - 100u\sin\theta + 2500 = 0\). [3]
2. 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]

A particle \(P\) is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t\) are denoted by \(x\) and \(y\) respectively.

1. Derive the equation of the trajectory of \(P\) in the form $$y = x\tan\alpha - \frac{gx^2}{2u^2}\sec^2\alpha.$$ [3]

The point \(Q\) is the highest point on the trajectory of \(P\) in the case where \(\alpha = 45°\).

1. Show that the \(x\)-coordinate of \(Q\) is \(\frac{u^2}{2g}\). [3]
2. Find the other value of \(\alpha\) for which \(P\) would pass through the point \(Q\). [4]

A particle \(P\) of mass \(m\) kg is held at rest at a point \(O\) and released so that it moves vertically under gravity against a resistive force of magnitude \(0.1mv^2\) N, where \(v\) m s\(^{-1}\) is the velocity of \(P\) at time \(t\) s.

1. Find an expression for \(v\) in terms of \(t\). [6]
2. Find an expression for \(v^2\) in terms of \(x\). [5]

The displacement of \(P\) from \(O\) at time \(t\) s is \(x\) m.

A particle \(P\) of mass \(mk\) falls from rest due to gravity. There is a resistance force of magnitude \(mkv^2\) N, where \(v\) ms\(^{-1}\) is the speed of \(P\) after it has fallen a distance \(x\) m and \(k\) is a positive constant.

1. By using \(v \frac{dv}{dx} = \frac{dv}{dt}\) and appropriate differential equation, show that $$v^2 = \frac{g}{k}(1 - e^{-2kx}).$$ [7] It is given that \(k = 0.01\). The speed of \(P\) when \(x = 0.2\) comes to approximately \(v\) ms\(^{-1}\).
2. 1. Find \(V\) correct to 2 decimal places. [1]
   2. Hence find how far \(P\) has fallen when its speed is \(\frac{1}{2}V\) ms\(^{-1}\). [2]

The points \(P\) and \(Q\) are at the same height \(h\) metres above horizontal ground. A small stone is dropped from rest from \(P\). Half a second later a second small stone is thrown vertically downwards from \(Q\) with speed 7.35 m s\(^{-1}\). Given that the stones hit the ground at the same time, find the value of \(h\). [7]

A small stone is projected vertically upwards from the point \(O\) and moves freely under gravity. The point \(A\) is 3.6 m vertically above \(O\). When the stone first reaches \(A\), the stone is moving upwards with speed 11.2 m s\(^{-1}\). The stone is modelled as a particle.

1. Find the maximum height above \(O\) reached by the stone. [4]
2. Find the total time between the instant when the stone was projected from \(O\) and the instant when it returns to \(O\). [5]
3. Sketch a velocity-time graph to represent the motion of the stone from the instant when it passes through \(A\) moving upwards to the instant when it returns to \(O\). Show, on the axes, the coordinates of the points where your graph meets the axes. [4]

A small ball is projected vertically upwards from a point \(O\) with speed 14.7 m s\(^{-1}\). The point \(O\) is 2.5 m above the ground. The motion of the ball is modelled as that of a particle moving freely under gravity. Find

1. the maximum height above the ground reached by the ball, [4]
2. the time taken for the ball to first reach a height of 1 m above the ground, [4]
3. the speed of the ball at the instant before it strikes the ground for the first time. [3]

A small ball is projected vertically upwards with speed \(29.4\text{ ms}^{-1}\) from a point \(A\) which is \(19.6\text{ m}\) above horizontal ground. The ball is modelled as a particle moving freely under gravity until it hits the ground. It is assumed that the ball does not rebound.

1. Find the distance travelled by the ball while its speed is less than \(14.7\text{ ms}^{-1}\) [3]
2. Find the time for which the ball is moving with a speed of more than \(29.4\text{ ms}^{-1}\) [3]
3. Sketch a speed-time graph for the motion of the ball from the instant when it is projected from \(A\) to the instant when it hits the ground. Show clearly where your graph meets the axes. [3]

A ball is projected vertically upwards with a speed of 14.7 m s\(^{-1}\) from a point which is 49 m above horizontal ground. Modelling the ball as a particle moving freely under gravity, find

1. the greatest height, above the ground, reached by the ball, [4]
2. the speed with which the ball first strikes the ground, [3]
3. the total time from when the ball is projected to when it first strikes the ground. [3]

\includegraphics{figure_3} Two particles \(A\) and \(B\) have mass 0.4 kg and 0.3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed above a horizontal floor. Both particles are held, with the string taut, at a height of 1 m above the floor, as shown in Figure 3. The particles are released from rest and in the subsequent motion \(B\) does not reach the pulley.

1. Find the tension in the string immediately after the particles are released. [6]
2. Find the acceleration of \(A\) immediately after the particles are released. [2]

When the particles have been moving for 0.5 s, the string breaks.

1. Find the further time that elapses until \(B\) hits the floor. [9]

A ball is projected vertically upwards with a speed \(u\) m s\(^{-1}\) from a point \(A\) which is 1.5 m above the ground. The ball moves freely under gravity until it reaches the ground. The greatest height attained by the ball is 25.6 m above \(A\).

1. Show that \(u = 22.4\). [3]

The ball reaches the ground 7 seconds after it has been projected from \(A\).

1. Find, to 2 decimal places, the value of \(T\). [4]

The ground is soft and the ball sinks 2.5 cm into the ground before coming to rest. The mass of the ball is 0.6 kg. The ground is assumed to exert a constant resistive force of magnitude \(F\) newtons.

1. Find, to 3 significant figures, the value of \(F\). [6]
2. State one physical factor which could be taken into account to make the model used in this question more realistic. [1]

A stone is thrown vertically upwards with speed \(16 \text{ m s}^{-1}\) from a point \(h\) metres above the ground. The stone hits the ground \(4\) s later. Find

1. the value of \(h\), [3]
2. the speed of the stone as it hits the ground. [3]

A ball is projected vertically upwards with speed 21 m s\(^{-1}\) from a point \(A\), which is 1.5 m above the ground. After projection, the ball moves freely under gravity until it reaches the ground. Modelling the ball as a particle, find

1. the greatest height above \(A\) reached by the ball, [3]
2. the speed of the ball as it reaches the ground, [3]
3. the time between the instant when the ball is projected from \(A\) and the instant when the ball reaches the ground. [4]

\includegraphics{figure_4} Figure 4 shows two particles \(P\) and \(Q\), of mass 3 kg and 2 kg respectively, connected by a light inextensible string. Initially \(P\) is held at rest on a fixed smooth plane inclined at 30° to the horizontal. The string passes over a small smooth light pulley \(A\) fixed at the top of the plane. The part of the string from \(P\) to \(A\) is parallel to a line of greatest slope of the plane. The particle \(Q\) hangs freely below \(A\). The system is released from rest with the string taut.

1. Write down an equation of motion for \(P\) and an equation of motion for \(Q\). [4]
2. Hence show that the acceleration of \(Q\) is 0.98 m s\(^{-2}\). [2]
3. Find the tension in the string. [2]
4. State where in your calculations you have used the information that the string is inextensible. [1]

On release, \(Q\) is at a height of 0.8 m above the ground. When \(Q\) reaches the ground, it is brought to rest immediately by the impact with the ground and does not rebound. The initial distance of \(P\) from \(A\) is such that in the subsequent motion \(P\) does not reach \(A\). Find

1. the speed of \(Q\) as it reaches the ground, [2]
2. the time between the instant when \(Q\) reaches the ground and the instant when the string becomes taut again. [5]

At time \(t = 0\) a ball is projected vertically upwards from a point \(O\) and rises to a maximum height of 40 m above \(O\). The ball is modelled as a particle moving freely under gravity.

1. Show that the speed of projection is 28 m s\(^{-1}\). [3]
2. Find the times, in seconds, when the ball is 33.6 m above \(O\). [5]

At time \(t = 0\), two balls \(A\) and \(B\) are projected vertically upwards. The ball \(A\) is projected vertically upwards with speed 2 m s\(^{-1}\) from a point 50 m above the horizontal ground. The ball \(B\) is projected vertically upwards from the ground with speed 20 m s\(^{-1}\). At time \(t = T\) seconds, the two balls are at the same vertical height, \(h\) metres, above the ground. The balls are modelled as particles moving freely under gravity. Find

1. the value of \(T\), [5]
2. the value of \(h\). [2]

A post is driven into the ground by means of a blow from a pile-driver. The pile-driver falls from rest from a height of \(1.6\) m above the top of the post.

1. Show that the speed of the pile-driver just before it hits the post is \(5.6\) m s\(^{-1}\). [2]

The post has mass \(6\) kg and the pile-driver has mass \(78\) kg. When the pile-driver hits the top of the post, it is assumed that the there is no rebound and that both then move together with the same speed.

1. Find the speed of the pile-driver and the post immediately after the pile-driver has hit the post. [3]

The post is brought to rest by the action of a resistive force from the ground acting for \(0.06\) s. By modelling this force as constant throughout this time,

1. find the magnitude of the resistive force, [4]
2. find, to 2 significant figures, the distance travelled by the post and the pile-driver before they come to rest. [4]

A small ball is projected vertically upwards from a point A. The greatest height reached by the ball is 40 m above A. Calculate

1. the speed of projection. [3]
2. the time between the instant that the ball is projected and the instant it returns to A. [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 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]

\includegraphics{figure_3} A rocket \(R\) of mass 100 kg is projected from a point \(A\) with speed 80 m s\(^{-1}\) at an angle of elevation of \(60°\), as shown in Fig. 3. The point \(A\) is 20 m vertically above a point \(O\) which is on horizontal ground. The rocket \(R\) moves freely under gravity. At \(B\) the velocity of \(R\) is horizontal. By modelling \(R\) as a particle, find

1. the height in m of \(B\) above the ground, [4]

1. the time taken for \(R\) to reach \(B\) from \(A\). [2]

When \(R\) is at \(B\), there is an internal explosion and \(R\) breaks into two parts \(P\) and \(Q\) of masses 60 kg and 40 kg respectively. Immediately after the explosion the velocity of \(P\) is 80 m s\(^{-1}\) horizontally away from \(A\). After the explosion the paths of \(P\) and \(Q\) remain in the plane \(OAB\). Part \(Q\) strikes the ground at \(C\). By modelling \(P\) and \(Q\) as particles,

1. show that the speed of \(Q\) immediately after the explosion is 20 m s\(^{-1}\), [3]

1. find the distance \(OC\). [6]

END

\includegraphics{figure_3} A rocket \(R\) of mass 100 kg is projected from a point \(A\) with speed 80 m s\(^{-1}\) at an angle of elevation of 60°, as shown in Fig. 3. The point \(A\) is 20 m vertically above a point \(O\) which is on horizontal ground. The rocket \(R\) moves freely under gravity. At \(B\) the velocity of \(R\) is horizontal. By modelling \(R\) as a particle, find

1. the height in m of \(B\) above the ground, [4]
2. the time taken for \(R\) to reach \(B\) from \(A\). [2]

When \(R\) is at \(B\), there is an internal explosion and \(R\) breaks into two parts \(P\) and \(Q\) of masses 60 kg and 40 kg respectively. Immediately after the explosion the velocity of \(P\) is 80 m s\(^{-1}\) horizontally away from \(A\). After the explosion the paths of \(P\) and \(Q\) remain in the plane \(OAB\). Part \(Q\) strikes the ground at \(C\). By modelling \(P\) and \(Q\) as particles,

1. show that the speed of \(Q\) immediately after the explosion is 20 m s\(^{-1}\), [3]
2. find the distance \(OC\). [6]