3.02d Constant acceleration: SUVAT formulae

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.

A cyclist starts from rest at point \(A\) and moves in a straight line with acceleration 0.5 m s\(^{-2}\) for a distance of 36 m. The cyclist then travels at constant speed for 25 s before slowing down, with constant deceleration, to come to rest at point \(B\). The distance \(AB\) is 210 m.

1. Find the total time that the cyclist takes to travel from \(A\) to \(B\). [5]
2. Find the time that it takes from when the cyclist starts until the car overtakes her. [5]

24 s after the cyclist leaves point \(A\), a car starts from rest from point \(A\), with constant acceleration 4 m s\(^{-2}\) towards \(B\). It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.

\includegraphics{figure_1} A small ball \(B\) is projected from a point \(O\) on horizontal ground towards a point \(A\) 12 m above the ground. 0.9 s after projection \(B\) has travelled a horizontal distance of 20 m and is vertically below \(A\) (see diagram).

1. Find the angle and the speed of projection of \(B\). [4]
2. Calculate the distance \(AB\) when \(B\) is vertically below \(A\). [2]

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 \(20 \text{ m s}^{-1}\) at an angle of \(60°\) below the horizontal, from a point \(O\) which is \(30 \text{ m}\) above horizontal ground.

1. Calculate the time taken by \(P\) to reach the ground. [3]
2. Calculate the speed and direction of motion of \(P\) immediately before it reaches the ground. [4]

\includegraphics{figure_7} A particle \(P\) is projected from a point \(O\) with initial speed \(10 \text{ m s}^{-1}\) at an angle of \(45°\) above the horizontal. \(P\) subsequently passes through the point \(A\) which is at an angle of elevation of \(30°\) from \(O\) (see diagram). At time \(t \text{ s}\) after projection the horizontal and vertically upward displacements of \(P\) from \(O\) are \(x \text{ m}\) and \(y \text{ m}\) respectively.

1. Write down expressions for \(x\) and \(y\) in terms of \(t\), and hence obtain the equation of the trajectory of \(P\). [3]
2. Calculate the value of \(x\) when \(P\) is at \(A\). [3]
3. Find the angle the trajectory makes with the horizontal when \(P\) is at \(A\). [4]

A particle \(P\) is projected with speed \(26\) m s\(^{-1}\) at an angle of \(30°\) above the horizontal from a point \(O\) on a horizontal plane.

1. For the instant when the vertical component of the velocity of \(P\) is \(5\) m s\(^{-1}\) downwards, find the direction of motion of \(P\) and the height of \(P\) above the plane. [4]
2. \(P\) strikes the plane at the point \(A\). Calculate the time taken by \(P\) to travel from \(O\) to \(A\) and the distance \(OA\). [3]

\includegraphics{figure_2} A particle \(P\) is projected from a point \(O\) at an angle of \(60°\) above horizontal ground. At an instant 0.6 s after projection, the angle of elevation of \(P\) from \(O\) is \(45°\) (see diagram).

1. Show that the speed of projection of \(P\) is 8.20 m s\(^{-1}\), correct to 3 significant figures. [4]
2. Calculate the time after projection when the direction of motion of \(P\) is \(45°\) above the horizontal. [3]

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

1. show that the vertical component of the velocity of \(P\) is \(8\) m s\(^{-1}\) downwards, [2]
2. calculate the distance of \(P\) from \(O\). [5]

A particle \(P\) is projected with speed \(V\text{ m s}^{-1}\) at an angle of \(60°\) above the horizontal from a point \(O\). At the instant \(1\text{ s}\) later a particle \(Q\) is projected from \(O\) with the same initial speed at an angle of \(45°\) above the horizontal. The two particles collide when \(Q\) has been in motion for \(t\text{ s}\).

1. Show that \(t = 2.414\), correct to 3 decimal places. [3]
2. Find the value of \(V\). [4]

The collision occurs after \(P\) has passed through the highest point of its trajectory.

1. Calculate the vertical distance of \(P\) below its greatest height when \(P\) and \(Q\) collide. [4]

A particle \(P\) is projected with speed \(V\,\text{m s}^{-1}\) at an angle of \(60°\) above the horizontal from a point \(O\). At the instant \(1\,\text{s}\) later a particle \(Q\) is projected from \(O\) with the same initial speed at an angle of \(45°\) above the horizontal. The two particles collide when \(Q\) has been in motion for \(t\,\text{s}\).

1. Show that \(t = 2.414\), correct to \(3\) decimal places. [3]
2. Find the value of \(V\). [4]

The collision occurs after \(P\) has passed through the highest point of its trajectory.

1. Calculate the vertical distance of \(P\) below its greatest height when \(P\) and \(Q\) collide. [4]

A particle \(P\) is projected with speed 35 m 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 \(x\) m and \(y\) m respectively. The equation of the trajectory of \(P\) is $$y = kx - \frac{(1 + k^2)x^2}{245},$$ where \(k\) is a constant. \(P\) passes through the points \(A(14, a)\) and \(B(42, 2a)\), 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°, correct to 3 decimal places. [5]

For the larger angle of projection, calculate

1. the time after projection when \(P\) passes through \(A\), [2]
2. the speed and direction of motion of \(P\) when it passes through \(B\). [4]

A small ball \(B\) is projected with speed \(30\text{ m s}^{-1}\) at an angle of \(60°\) to the horizontal from a point on horizontal ground. Find the time after projection when the speed of \(B\) is \(25\text{ m s}^{-1}\) for the second time. [4]

\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 \(t\) s after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x\) m and \(y\) 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 particle \(P\) is projected from a point \(O\) on a horizontal plane and moves freely under gravity. The initial velocity of \(P\) is 100 ms\(^{-1}\) at an angle \(\theta\) above the horizontal, where \(\tan\theta = \frac{4}{3}\). The two times at which \(P\)'s height above the plane is \(H\) m differ by 10 s.

1. Find the value of \(H\). [5]

1. Find the magnitude and direction of the velocity of \(P\) one second before it strikes the plane. [4]

Particles \(P\) and \(Q\) are projected in the same vertical plane from a point \(O\) at the top of a cliff. The height of the cliff exceeds 50 m. Both particles move freely under gravity. Particle \(P\) is projected with speed \(\frac{35}{2} \text{ m s}^{-1}\) at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac{4}{3}\). Particle \(Q\) is projected with speed \(u \text{ m s}^{-1}\) at an angle \(\beta\) above the horizontal, where \(\tan \beta = \frac{1}{2}\). Particle \(Q\) is projected one second after the projection of particle \(P\). The particles collide \(T\) s after the projection of particle \(Q\).

1. Write down expressions, in terms of \(T\), for the horizontal displacements of \(P\) and \(Q\) from \(O\) when they collide and hence show that \(4uT = 21\sqrt{5(T + 1)}\). [4]
2. Find the value of \(T\). [4]
3. Find the horizontal and vertical displacements of the particles from \(O\) when they collide. [3]

At time \(t\)s, a particle \(P\) is projected with speed \(40\)m s\(^{-1}\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The greatest height achieved by \(P\) during its flight is \(H\)m and the corresponding time is \(T\)s.

1. Obtain expressions for \(H\) and \(T\) in terms of \(\theta\). [2]

During the time between \(t = T\) and \(t = 3\), \(P\) descends a distance \(\frac{1}{4}H\).

1. Find the value of \(\theta\). [4]
2. Find the speed of \(P\) when \(t = 3\). [3]

The points \(O\) and \(P\) are on a horizontal plane, a distance \(8\) m apart. A ball is thrown from \(O\) with speed \(u\) m s\(^{-1}\) at an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac{3}{4}\). At the same instant, a model aircraft is launched with speed \(5\) m s\(^{-1}\) parallel to the horizontal plane from a point \(4\) m vertically above \(P\). The model aircraft moves in the same vertical plane as the ball and in the same horizontal direction as the ball. The model aircraft moves horizontally with a constant speed of \(5\) m s\(^{-1}\). After \(T\) s, the ball and the model aircraft collide.

1. Find the value of \(T\). [6]
2. Find the direction in which the ball is moving immediately before the collision. [3]

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 train travels along a straight horizontal track between two stations \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration until it reaches its maximum speed of 108 km h\(^{-1}\). The train then travels at this speed before it moves with constant deceleration coming to rest at \(B\). The journey from \(A\) to \(B\) takes 8 minutes.

1. Change 108 km h\(^{-1}\) into m s\(^{-1}\). [2]
2. Sketch a speed-time graph for the motion of the train between the two stations \(A\) and \(B\). [2]

Given that the distance between the two stations is 12 km and that the time spent decelerating is three times the time spent accelerating,

1. find the acceleration, in m s\(^{-2}\), of the train. [6]

\includegraphics{figure_3} A particle \(A\) of mass \(3m\) is held at rest on a rough horizontal table. The particle is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) which is fixed at the edge of the table. The other end of the string is attached to a particle \(B\) of mass \(2m\), which hangs freely, vertically below \(P\). The system is released from rest, with the string taut, when \(A\) is 1.3 m from \(P\) and \(B\) is 1 m above the horizontal floor, as shown in Figure 3. Given that \(B\) hits the floor 2 s after release and does not rebound,

1. find the acceleration of \(A\) during the first two seconds, [2]
2. find the coefficient of friction between \(A\) and the table, [8]
3. determine whether \(A\) reaches the pulley. [6]

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]

\includegraphics{figure_3} A particle \(P\) of mass 2 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass 5 kg is attached to the other end of the string. The string passes over a small smooth light pulley. The pulley is fixed at a point on the intersection of a rough horizontal table and a fixed smooth inclined plane. The string lies along the table and also lies in a vertical plane which contains a line of greatest slope of the inclined plane. This plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\). Particle \(P\) is at rest on the table, a distance \(d\) metres from the pulley. Particle \(Q\) is on the inclined plane with the string taut, as shown in Figure 3. The coefficient of friction between \(P\) and the table is \(\frac{1}{4}\). The system is released from rest and \(P\) slides along the table towards the pulley. Assuming that \(P\) has not reached the pulley and that \(Q\) remains on the inclined plane,

1. write down an equation of motion for \(P\), [2]
2. write down an equation of motion for \(Q\), [2]
3. 1. find the acceleration of \(P\),
   2. find the tension in the string. [5]

When \(P\) has moved a distance 0.5 m from its initial position, the string breaks. Given that \(P\) comes to rest just as it reaches the pulley,

1. find the value of \(d\). [7]

A car is moving along a straight horizontal road with constant acceleration \(a\) m s\(^{-2}\) (\(a > 0\)). At time \(t = 0\) the car passes the point \(P\) moving with speed \(u\) m s\(^{-1}\). In the next 4 s, the car travels 76 m and then in the following 6 s it travels a further 219 m. Find

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

[7]

A block \(A\) of mass 9 kg is released from rest from a point \(P\) which is a height \(h\) metres above horizontal soft ground. The block falls and strikes another block \(B\) of mass 1.5 kg which is on the ground vertically below \(P\). The speed of \(A\) immediately before it strikes \(B\) is 7 m s\(^{-1}\). The blocks are modelled as particles.

1. Find the value of \(h\). [2] Immediately after the impact the blocks move downwards together with the same speed and both come to rest after sinking a vertical distance of 12 cm into the ground. Assuming that the resistance offered by the ground has constant magnitude \(R\) newtons,
2. find the value of \(R\). [8]