Projectiles

A small stone is projected with speed 14 m s^–1 from a point O on the top of a cliff. The point O is H metres vertically above the point N. Point N is on horizontal ground. The stone is projected at an angle θ above the horizontal, where tanθ = The stone strikes the horizontal ground at the point A, where NA = 40 m, as shown in Figure 3. The stone is modelled as a particle moving freely under gravity. Using this model, find

the value of H

the maximum height of the stone above the horizontal ground.

At time t = 0, a small stone is projected with velocity 35 m s^–1 from a point O on horizontal ground. The stone is projected at an angle α to the horizontal, where tanα = In an initial model

• the stone is modelled as a particle P moving freely under gravity

• the stone hits the ground at the point A

Figure 4 shows the path of P from O to A. For the motion of P from O to A

• at time t seconds, the horizontal distance of P from O is x metres

• at time t seconds, the vertical distance of P above the ground is y metres

Using the model, show that

Use the answer to (a), or otherwise, to find the length OA. Using the model, the greatest height of the stone above the ground is found to be H metres.

Use the answer to (a), or otherwise, to find the value of H.

• The model is refined to include air resistance.

Using this new model, the greatest height of the stone above the ground is found to be K metres.

State which is greater, H or K, justifying your answer.

State one limitation of this refined model.

A small ball is projected with speed 28 m s^–1 from a point O on horizontal ground. After moving for T seconds, the ball passes through the point A. The point A is 40 m horizontally and 20 m vertically from the point O, as shown in Figure 2. The motion of the ball from O to A is modelled as that of a particle moving freely under gravity. Given that the ball is projected at an angle α to the ground, use the model to

show that tan²α – 4 tanα + 3 = 0

find the greatest possible height, in metres, of the ball above the ground as the ball moves from O to A. The model does not include air resistance.

State one other limitation of the model. _____________________________________________________________________________________

A small stone is projected with speed 65 m s⁻¹ from a point O at the top of a vertical cliff.

Point O is 70 m vertically above the point N.

Point N is on horizontal ground. The stone is projected at an angle α above the horizontal, where tanα =

The stone hits the ground at the point A, as shown in Figure 3.

The stone is modelled as a particle moving freely under gravity.

The acceleration due to gravity is modelled as having magnitude 10 m s⁻²

Using the model,

find the time taken for the stone to travel from O to A,

find the speed of the stone at the instant just before it hits the ground at A.

One limitation of the model is that it ignores air resistance.

State one other limitation of the model that could affect the reliability of your answers.

A small stone is projected with speed 65 m s⁻¹ from a point O at the top of a vertical cliff. Point O is 70 m vertically above the point N. Point N is on horizontal ground. The stone is projected at an angle α above the horizontal, where tanα = The stone hits the ground at the point A, as shown in Figure 3. The stone is modelled as a particle moving freely under gravity. The acceleration due to gravity is modelled as having magnitude 10 m s⁻²

Using the model,

find the time taken for the stone to travel from O to A,

find the speed of the stone at the instant just before it hits the ground at A. One limitation of the model is that it ignores air resistance.

State one other limitation of the model that could affect the reliability of your answers.

A small ball is projected with speed Um s⁻¹ from a point O at the top of a vertical cliff. The point O is 25 m vertically above the point N which is on horizontal ground. The ball is projected at an angle of 45° above the horizontal. The ball hits the ground at a point A, where AN = 100 m, as shown in Figure 2. The motion of the ball is modelled as that of a particle moving freely under gravity. Using this initial model,

show that U = 28

find the greatest height of the ball above the horizontal ground NA. In a refinement to the model of the motion of the ball from O to A, the effect of air resistance is included. This refined model is used to find a new value of U.

How would this new value of U compare with 28, the value given in part (a)?

State one further refinement to the model that would make the model more realistic.

TOTAL FOR MECHANICS IS 50 MARKS

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 α above the horizontal, where tan = 2, and moves freely under gravity. When the particle has moved a horizontal distance x from O, its height above the ground is y.

Show that

The particle hits the ground at the point A, where OA = 36m.

Find u, the speed of projection.

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.

Find the horizontal distance between O and B.

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 AB and move freely under gravity. Particle P is projected from A with speed 30 m s^–1 at 60 to AB and particle Q is projected from B with speed q m s^–1 at angle to BA, as shown in Figure 4. At t = 2 seconds, P and Q collide.

Find

the value of q.

Find the speed of P at the instant before it collides with Q.

[ In this question, i is a horizontal unit vector and j is an upward vertical unit vector.] A particle P is projected from a fixed origin O with velocity (3i + 4j)m s. The particle moves freely under gravity and passes through the point A with position vector (i – j)m, where is a positive constant.

Find

the speed of P at the instant when it passes through A,

the direction of motion of P at the instant when it passes through A.

A small ball is projected with speed 14 m s^–1 from a point A on horizontal ground. The angle of projection is 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 m s^–1.

Find the value of h.

find the horizontal distance from A to B.

A particle P is projected from a point A with speed 25 m s^–1 at an angle of elevation , where sin α . 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

the greatest height above the ground of P, as it moves from A to B,

the distance OB.

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.

Find the time taken by P to move from A to C.

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 m s^–1 at an angle 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 2u m s^–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

the minimum speed of the ball on its path from A to B.

A small ball is projected from a fixed point O so as to hit a target T which is at a horizontal distance 9 a from O and at a height 6 a above the level of O. The ball is projected with speed (27ag) at an angle θ to the horizontal, as shown in Figure 4. The ball is modelled as a particle moving freely under gravity.

Show that tan² – 6 tan θ + 5 = 0

The two possible angles of projection are ₁ and ₂, where ₁ > ₂.

Find tan ₁ and tan ₂.

The particle is projected at the larger angle ₁.

Find the speed of the particle immediately before it hits T.

A ball is thrown from a point O, which is 6 m above horizontal ground. The ball is projected with speed u m s⁻¹ at an angle 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.

Find the value of u.

The ball hits the ground T seconds after projection.

Find the value of T.

Immediately before the ball hits the ground the direction of motion of the ball makes an angle with the horizontal.

A small stone is projected from a point O at the top of a vertical cliff OA. 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 AB = 50 m, as shown in Figure 4. The stone is modelled as a particle moving freely under gravity.

Show that the vertical component of the velocity of projection of the stone is 14 m s^–1.

Find the speed of projection.

Find the time after projection when the stone is moving parallel to OB.

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

Show that

A girl throws a ball from a point A at the top of a cliff. The point A is 8 m above a horizontal beach. The ball is projected with speed 7 m s¹ at an angle of elevation of 45°. By modelling the ball as a particle moving freely under gravity,

find the horizontal distance of the ball from A when the ball is 1 m above the beach.

A boy is standing on the beach at the point B vertically below A. He starts to run in a straight line with speed v m s¹, leaving B 0.4 seconds after the ball is thrown. He catches the ball when it is 1 m above the beach.

Find the value of v.

[ In this question, the unit vectors i and j are in a vertical plane, i being horizontal and j being vertically upwards. ]

At time t = 0, a particle P is projected from the point A which has position vector 10 j metres with respect to a fixed origin O at ground level. The ground is horizontal. The velocity of projection of P is (3i + 5j) m s^–1, as shown in Figure 3. The particle moves freely under gravity and reaches the ground after T seconds.

For 0 tT, show that, with respect to O, the position vector, r metres, of P at time t seconds is given by

Find the value of T.

Find the velocity of P at time t seconds (0 t T).

When P is at the point B, the direction of motion of P is 45° below the horizontal.

Find the time taken for P to move from A to B.

Find the speed of P as it passes through B.

A ball is projected with speed 40 m 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 θ ° 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

the value of θ,

the distance OR,

the speed of the ball as it hits the ground at R.

[ In this question i and j are unit vectors in a horizontal and upward vertical direction respectively ] A particle P is projected from a fixed point O on horizontal ground with velocity u(i + cj) m s^–1, where c and u are positive constants. The particle moves freely under gravity until it strikes the ground at A, where it immediately comes to rest. Relative to O, the position vector of a point on the path of P is (x i + y j) m.

Show that

Given that u = 7, OA = R m and the maximum vertical height of P above the ground is H m,

using the result in part (a), or otherwise, find, in terms of c,

R

H.

Given also that when P is at the point Q, the velocity of P is at right angles to its initial velocity,

find, in terms of c, the value of x at Q.

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 m s^–1 from point O on the ground at an angle α to the ground.

By writing down expressions for the horizontal and vertical distances, from O of the ball t seconds after it was hit, show that

Given that α = 45°,

find the speed of the ball as it passes over the fence.

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 m s^–1 at an angle of 30° below the horizontal. The ball is modelled as a particle and the target as a point T. The distance OT is 15 m. The ball misses the target and hits the ground at the point B, where OTB is a straight line, as shown in Figure 4. Find

the time taken by the ball to travel from A to B,

the distance TB.

The point X is on the path of the ball vertically above T.

Find the speed of the ball at X.

[ In this question, the unit vectors i and j are in a vertical plane, i being horizontal and j being vertical. ] A particle P is projected from the point A which has position vector 47.5 j metres with respect to a fixed origin O. The velocity of projection of P is (2ui + 5uj) m s^–1. The particle moves freely under gravity passing through the point B with position vector 30 i metres, as shown in Figure 3.

Show that the time taken for P to move from A to B is 5 s.

Find the value of u.

Find the speed of P at B.