Deriving trajectory equation

A stone is projected from a point \(O\) on horizontal ground. The equation of the trajectory of the stone is $$y = 1.2x - 0.15x^2,$$ where \(x\) m and \(y\) m are respectively the horizontal and vertically upwards displacements of the stone from \(O\). Find

1. the greatest height of the stone, [2]
2. the distance from \(O\) of the point where the stone strikes the ground. [2]

A particle is projected from a point \(O\) on horizontal ground. The initial components of the velocity of the particle are \(10\,\text{m}\,\text{s}^{-1}\) horizontally and \(15\,\text{m}\,\text{s}^{-1}\) vertically. At time \(t\) s after projection, the horizontal and vertically upwards displacements of the particle from \(O\) are \(x\) m and \(y\) m respectively.

1. Express \(x\) and \(y\) in terms of \(t\), and hence find the equation of the trajectory of the particle. [4]
2. The horizontal ground is at the top of a vertical cliff. The point \(O\) is at a distance \(d\) m from the edge of the cliff. The particle is projected towards the edge of the cliff and does not strike the ground before it passes over the edge of the cliff. Show that \(d\) is less than \(30\). [2]
3. Find the value of \(x\) when the particle is \(14\) m below the level of \(O\). [2]

A particle is projected from a point \(O\) on horizontal ground. The initial components of the velocity of the particle are \(10 \text{ ms}^{-1}\) horizontally and \(15 \text{ ms}^{-1}\) vertically. At time \(t\) s after projection, the horizontal and vertically upwards displacements of the particle from \(O\) are \(x\) m and \(y\) m respectively.

1. Express \(x\) and \(y\) in terms of \(t\), and hence find the equation of the trajectory of the particle. [4]
2. Show that \(d\) is less than \(30\). [2]
3. Find the value of \(x\) when the particle is \(14\) m below the level of \(O\). [2]

The horizontal ground is at the top of a vertical cliff. The point \(O\) is at a distance \(d\) m from the edge of the cliff. The particle is projected towards the edge of the cliff and does not strike the ground before it passes over the edge of the cliff.

A particle \(P\) is projected from a point \(O\) on horizontal ground with initial speed 20 m s\(^{-1}\) and angle of projection 30°. At time \(t\) s after projection, the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x\) m and \(y\) m respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of \(P\). [4]
2. Calculate this height. [3]

\(P\) is at the same height above the ground at two points which are a horizontal distance apart of 15 m.

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 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]
2. The greatest height of \(P\) above the plane is denoted by \(H\). When \(P\) is at a height of \(\frac{3}{4}H\), it is travelling at a horizontal distance \(d\). Given that \(\tan \alpha = 3\) and in terms of \(H\), the two possible values of \(d\). [6]

A particle is projected from a point with speed \(u\) at an angle of elevation \(α\) above the horizontal and moves freely under gravity. When it has moved a horizontal distance \(x\), its height above the point of projection is \(y\).

1. Show that $$y = x \tan α - \frac{gx^2}{2u^2}(1 + \tan^2 α).$$ [5]

A shot-putter puts a shot from a point \(A\) at a height of 2 m above horizontal ground. The shot is projected at an angle of elevation of \(45°\) with a speed of 14 m s\(^{-1}\). By modelling the shot as a particle moving freely under gravity,

1. find, to 3 significant figures, the horizontal distance of the shot from \(A\) when the shot hits the ground, [5]

1. find, to 2 significant figures, the time taken by the shot in moving from \(A\) to reach the ground. [2]

A small smooth ball \(A\) of mass \(m\) is moving on a horizontal table with speed \(v\) when it collides directly with another small smooth ball \(B\) of mass \(3m\) which is at rest on the table. The balls have the same radius and the coefficient of restitution between the balls is \(e\). The direction of motion of \(A\) is reversed as a result of the collision.

1. Find, in terms of \(e\) and \(u\), the speeds of \(A\) and \(B\) immediately after the collision. [7]

In the subsequent motion \(B\) strikes a vertical wall, which is perpendicular to the direction of motion of \(B\), and rebounds. The coefficient of restitution between \(B\) and the wall is \(\frac{1}{2}\). Given that there is a second collision between \(A\) and \(B\),

1. find the range of values of \(e\) for which the motion described is possible. [6]

A particle is projected from a point with speed \(u\) at an angle of elevation \(\alpha\) above the horizontal and moves freely under gravity. When it has moved a horizontal distance \(x\), its height above the point of projection is \(y\).

1. Show that $$y = x \tan \alpha - \frac{gx^2}{2u^2}(1 + \tan^2 \alpha).$$ [5]

A shot-putter puts a shot from a point \(A\) at a height of 2 m above horizontal ground. The shot is projected at an angle of elevation of 45° with a speed of 14 m s\(^{-1}\). By modelling the shot as a particle moving freely under gravity,

1. find, to 3 significant figures, the horizontal distance of the shot from \(A\) when the shot hits the ground, [5]
2. find, to 2 significant figures, the time taken by the shot in moving from \(A\) to reach the ground. [2]

\includegraphics{figure_3} Fig. 7 shows the graph of \(y = \frac{1}{100}(100 + 15x - x^2)\). For \(0 \leq x < 20\), this graph shows the trajectory of a small stone projected from the point Q where \(y\) m is the height of the stone above horizontal ground and \(x\) m is the horizontal displacement of the stone from O. The stone hits the ground at the point R.

1. Write down the height of Q above the ground. [1]
2. Find the horizontal distance from O of the highest point of the trajectory and show that this point is \(1.5625\) m above the ground. [5]
3. Show that the time taken for the stone to fall from its highest point to the ground is \(0.565\) seconds, correct to 3 significant figures. [3]
4. Show that the horizontal component of the velocity of the stone is \(22.1\text{ms}^{-1}\), correct to 3 significant figures. Deduce the time of flight from Q to R. [5]
5. Calculate the speed at which the stone hits the ground. [4]

Take \(g = 10\) ms\(^{-2}\) in this question. \includegraphics{figure_6} A golfer hits a ball from a point \(T\) at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{5}{13}\), giving it an initial speed of 52 ms\(^{-1}\). The ball lands on top of a mound, 15 m above the level of \(T\), as shown.

1. Show that the height, \(y\) m, of the ball above \(T\) at time \(t\) seconds after it was hit is given by $$y = 20t - 5t^2.$$ [3 marks]
2. Find the time for which the ball is in flight. [4 marks]
3. Find the horizontal distance travelled by the ball. [3 marks]
4. Show that, if the ball is \(x\) m horizontally from \(T\) at time \(t\) seconds, then $$y = \frac{5}{12}x - \frac{5}{2304}x^2.$$ [3 marks]
5. Name a force that has been ignored in your mathematical model and state whether the answer to part (b) would be larger or smaller if this force were taken into account. [2 marks]

A particle P is projected from a fixed point O with initial velocity \(u\mathbf{i} + ku\mathbf{j}\), where \(k\) is a positive constant. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the horizontal and vertically upward directions respectively. P moves with constant gravitational acceleration of magnitude \(g\). At time \(t \geq 0\), particle P has position vector \(\mathbf{r}\) relative to O.

1. Starting from an expression for \(\ddot{\mathbf{r}}\), use integration to derive the formula $$\mathbf{r} = ut\mathbf{i} + \left(kut - \frac{1}{2}gt^2\right)\mathbf{j}.$$ [4]

The position vector \(\mathbf{r}\) of P at time \(t \geq 0\) can be expressed as \(\mathbf{r} = x\mathbf{i} + y\mathbf{j}\), where the axes Ox and Oy are horizontally and vertically upwards through O respectively. The axis Ox lies on horizontal ground.

1. Show that the path of P has cartesian equation $$gy^2 - 2ku^2x + 2u^2y = 0.$$ [3]
2. Hence find, in terms of \(g\), \(k\) and \(u\), the maximum height of P above the ground during its motion. [3]

The maximum height P reaches above the ground is equal to the distance OA, where A is the point where P first hits the ground.

1. Determine the value of \(k\). [3]