Projectiles

A marble rolls off the edge of a horizontal table top with speed 2.5 m s⁻¹. The table top is 0.98 m above a horizontal floor. The marble is modelled as a particle moving freely under gravity from the moment it leaves the table. Take g = 9.8 m s⁻².

A pebble is flicked from a point O on horizontal ground with speed 12 m s⁻¹ at an angle α above the horizontal, where tan α = ⁴/₃. The pebble is modelled as a particle moving freely under gravity. Take g = 9.8 m s⁻².

A beanbag is thrown from a point on horizontal ground with speed 14 m s⁻¹ at an angle of 60° above the horizontal. The beanbag is modelled as a particle moving freely under gravity. Take g = 9.8 m s⁻².

A particle P is projected from a fixed point O with velocity (6i + 8j) m s⁻¹, where i and j are unit vectors horizontally and vertically upwards respectively. P moves freely under gravity. Take g = 9.8 m s⁻².

Parcels leave the end E of a horizontal conveyor belt with the speed of the belt and fall into an open bin on a horizontal floor. The diagram shows the dimensions: E is 1.6 m above the floor, and the open top of the bin is 0.375 m above the floor and stretches from 0.9 m to 1.5 m horizontally from E. Each parcel is modelled as a particle moving freely under gravity after it leaves E, and the sides of the bin have negligible thickness. A parcel lands in the bin if, when it has fallen to the level of the top of the bin, its horizontal distance from E is between 0.9 m and 1.5 m. Take g = 9.8 m s⁻².

A ball is struck from a point O on horizontal ground. It moves freely under gravity and lands on the ground 3 s later. The graph shows how v m s⁻¹, the vertical component of the velocity of the ball (upwards positive), varies with the time t seconds after the ball is struck. The horizontal component of the velocity is constant at 8 m s⁻¹.

A particle P is projected from a point O on horizontal ground with velocity (ai + bj) m s⁻¹, where i and j are unit vectors horizontally and vertically upwards respectively. P moves freely under gravity and lands at the point L on the ground. The diagram shows the path of P on a grid, with distances in metres. A and B are the positions of P 1 s and 2 s after projection, and their coordinates relative to O are shown. Take g = 9.8 m s⁻².

Two jets of water, A and B, leave a nozzle N at ground level in a garden. The diagram shows their paths, drawn to scale on a grid. Both jets reach the same greatest height, 3.6 m, and they land on the horizontal ground 6 m and 12 m from N respectively, at the grid points shown. Each drop of water is modelled as a particle moving freely under gravity. Take g = 9.8 m s⁻².

A ball is thrown horizontally from an open window. The point of release is 12.5 m vertically above horizontal ground. The ball hits the ground at a point which is 18 m horizontally from the point of release. The ball is modelled as a particle moving freely under gravity. Take g = 9.8 m s⁻².

A stone is thrown from a point A on a bridge. A is 22 m vertically above the surface of a river. The stone is thrown with speed 8 m s⁻¹ at an angle of 20° below the horizontal. The stone moves freely under gravity until it hits the water. Take g = 9.8 m s⁻².

In a game, a dart is thrown horizontally towards a board. The dart is released at a horizontal distance of 2.4 m from the board, level with the centre of the board, and it hits the board at a point 0.12 m vertically below the centre. The dart is modelled as a particle moving freely under gravity. Take g = 9.8 m s⁻².

A machine on horizontal ground fires tennis balls with speed 16 m s⁻¹ at an angle θ above the horizontal. The balls are modelled as particles moving freely under gravity, and the balls leave the machine at ground level. Take g = 9.8 m s⁻².

A ball is thrown from a point O with speed 13 m s⁻¹ at an angle θ above the horizontal, where tan θ = ¹²/₅. The ball moves freely under gravity in a vertical plane. Take O as the origin, with x m and y m the horizontal and vertically upward displacements of the ball from O. Take g = 9.8 m s⁻².

A fountain nozzle at ground level projects drops of water with speed 11 m s⁻¹ at an angle of 70° above the horizontal. Each drop is modelled as a particle moving freely under gravity. Take g = 9.8 m s⁻².

A ball is kicked from a point O on horizontal ground with speed u m s⁻¹ at an angle θ above the horizontal, and moves freely under gravity. The graph shows the path of the ball, where x m and y m are its horizontal and vertical distances from O. The highest point of the path is at a grid point, and the ball lands at x = 20. A vertical wall of height 3 m stands at x = 16. Take g = 9.8 m s⁻².

A ball is kicked from a point O on horizontal ground with speed u m s⁻¹ at 30° above the horizontal. It moves freely under gravity and lands at the point T on the flat top of a platform. The diagram shows the dimensions: the platform is 1.5 m high, its front edge is 15 m horizontally from O, and T is 18 m horizontally from O. Take g = 9.8 m s⁻².

A ball is projected from a point O on horizontal ground and moves freely under gravity. The diagram shows its path. At the point Q, 1.5 s after projection, the ball is moving with speed 10 m s⁻¹ in the direction shown, 30° below the horizontal. Take g = 9.8 m s⁻².

A ball is thrown from a point A, 2.1 m above horizontal ground, and moves freely under gravity until it hits the ground. The graphs show x m, the horizontal displacement of the ball from A, and y m, the height of the ball above the ground, against the time t seconds after it is thrown. The labelled points give exact values. Take g = 9.8 m s⁻².

A ball is thrown from the edge E of the flat roof of building A with speed 12 m s⁻¹ at 25° above the horizontal. The diagram shows the dimensions: E is 20 m above horizontal ground, and building B has a flat roof 8 m above the ground that begins 10 m horizontally from E and is 15 m long. The ball moves freely under gravity in the vertical plane shown. Take g = 9.8 m s⁻².

A stunt rider leaves the lip L of a ramp with speed u m s⁻¹ at 20° above the horizontal. The diagram shows the landing deck, whose horizontal top is level with L. The near end of the deck is 9 m horizontally from L and the deck is 6 m long. The rider and bike are modelled as a particle moving freely under gravity. Take g = 9.8 m s⁻².

A water balloon is launched from a point O with speed 24 m s⁻¹ at an angle θ above the horizontal. It passes through an open window W. The point W is 36 m horizontally from O and 8.5 m vertically above the level of O. The balloon is modelled as a particle moving freely under gravity. Take g = 9.8 m s⁻².

A particle P is projected from a point A with velocity (8i + 10j) m s⁻¹, where i and j are unit vectors horizontally and vertically upwards respectively. The point O is on horizontal ground and the position vector of A relative to O is (2i + 3j) m. P moves freely under gravity until it hits the ground. Take g = 9.8 m s⁻².

A stone is catapulted from a point C on the top of a vertical castle wall. C is 30 m above horizontal ground. The stone is projected with speed 24 m s⁻¹ at an angle of 40° above the horizontal and moves freely under gravity until it hits the ground. Take g = 9.8 m s⁻².

A particle is projected from a point O on horizontal ground with speed u m s⁻¹ at an angle θ above the horizontal. It moves freely under gravity, where the acceleration due to gravity is g m s⁻².

A ball is projected from a point O on horizontal ground with speed 15 m s⁻¹ at an angle θ above the horizontal, and moves freely under gravity. The diagram shows a vertical post standing on the ground; its top P is 12 m horizontally from O and 4 m above the ground. The ball must pass above P. Take g = 9.8 m s⁻².

The diagram shows a point O on horizontal ground and a vertical tower whose foot F is 20 m from O; the top T of the tower is 15 m above F. At time t = 0, a particle A is projected from O with velocity (10i + 15j) m s⁻¹, and at the same instant a particle B is projected horizontally from T towards O with speed V m s⁻¹. Here i and j are unit vectors horizontally (in the direction from O to F) and vertically upwards. Both particles move freely under gravity in the vertical plane through O, F and T. Take g = 9.8 m s⁻².

A ball is projected from a point A at the top of a vertical cliff with speed u m s⁻¹ at an angle α above the horizontal, where tan α = ³/₄. The ball moves freely under gravity and hits the sea at the point B, 5 s after projection. The diagram shows that A is h m above the sea and that B is 60 m horizontally from A. Take g = 9.8 m s⁻².

In a sports hall, a ball is launched from a point O on the floor with speed 21 m s⁻¹ at an angle θ above the horizontal. It moves freely under gravity and is aimed at the point P on the edge of a balcony. The diagram shows that P is 30 m horizontally from O and 5 m above the floor, and that the horizontal ceiling is 12 m above the floor. Take g = 9.8 m s⁻².