3.02d Constant acceleration: SUVAT formulae

4 \includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-2_522_959_1692_593} A sprinter runs a race of 400 m . His total time for running the race is 52 s . The diagram shows the velocity-time graph for the motion of the sprinter. He starts from rest and accelerates uniformly to a speed of \(8.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 6 s . The sprinter maintains a speed of \(8.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 36 s , and he then decelerates uniformly to a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the end of the race.

1. Calculate the distance covered by the sprinter in the first 42 s of the race.
2. Show that \(V = 7.84\).
3. Calculate the deceleration of the sprinter in the last 10 s of the race.

2 Alan starts walking from a point \(O\), at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along a horizontal path. Ben walks along the same path, also starting from \(O\). Ben starts from rest 5 s after Alan and accelerates at \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 5 s . Ben then continues to walk at a constant speed until he is at the same point, \(P\), as Alan.

1. Find how far Ben has travelled when he has been walking for 5 s and find his speed at this instant.
2. Find the distance \(O P\).

5 The motion of a car of mass 1400 kg is resisted by a constant force of magnitude 650 N .

1. Find the constant speed of the car on a horizontal road, assuming that the engine works at a rate of 20 kW .
2. The car is travelling at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 7 }\). Find the power of the car's engine.
3. The car descends the same hill with the engine working at \(80 \%\) of the power found in part (ii). Find the acceleration of the car at an instant when the speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6 Two particles of masses 1.3 kg and 0.7 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The particles are held at the same vertical height with the string taut. The distance of each particle above a horizontal plane is 2 m , and the distance of each particle below the pulley is 4 m . The particles are released from rest.

1. Find
   1. the tension in the string before the particle of mass 1.3 kg reaches the plane,
   2. the time taken for the particle of mass 1.3 kg to reach the plane.
   3. Find the greatest height of the particle of mass 0.7 kg above the plane.

2 A particle of mass 0.8 kg is projected with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a rough plane inclined at an angle of \(10 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.4 .

1. Find the acceleration of the particle.
2. Find the distance the particle moves up the plane before coming to rest.

5 A particle \(P\) moves in a straight line \(A B C D\) with constant deceleration. The velocities of \(P\) at \(A , B\) and \(C\) are \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 } , 12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.

1. Find the ratio of distances \(A B : B C\).
2. The particle comes to rest at \(D\). Given that the distance \(A D\) is 80 m , find the distance \(B C\).

3 A train travels between two stations, \(A\) and \(B\). The train starts from rest at \(A\) and accelerates at a constant rate for \(T\) s until it reaches a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then travels at this constant speed before decelerating at a constant rate, coming to rest at \(B\). The magnitude of the train's deceleration is twice the magnitude of its acceleration. The total time taken for the journey is 180 s .

1. Sketch the velocity-time graph for the train's journey from \(A\) to \(B\). \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-04_496_857_516_685}
2. Find an expression, in terms of \(T\), for the length of time for which the train is travelling with constant speed.
3. The distance from \(A\) to \(B\) is 3300 m . Find how far the train travels while it is decelerating.

5 A particle is projected vertically upwards from a point \(O\) with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Two seconds later a second particle is projected vertically upwards from \(O\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after the second particle is projected, the two particles collide.

1. Find \(t\). \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-06_65_1569_488_328}
2. Hence find the height above \(O\) at which the particles collide.

4 A particle \(P\) moves in a straight line \(A B C D\) with constant acceleration. The distances \(A B\) and \(B C\) are 100 m and 148 m respectively. The particle takes 4 s to travel from \(A\) to \(B\) and also takes 4 s to travel from \(B\) to \(C\).

1. Show that the acceleration of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the speed of \(P\) at \(A\).
2. \(P\) reaches \(D\) with a speed of \(61 \mathrm {~ms} ^ { - 1 }\). Find the distance \(C D\).

1 A bus moves in a straight line between two bus stops. The bus starts from rest and accelerates at \(2.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 5 s . The bus then travels for 24 s at constant speed and finally slows down, with a constant deceleration, stopping in a further 6 s . Sketch a velocity-time graph for the motion and hence find the distance between the two bus stops.

3 A car of mass 1250 kg travels down a straight hill with the engine working at a power of 22 kW . The hill is inclined at \(3 ^ { \circ }\) to the horizontal and the resistance to motion of the car is 1130 N . Find the speed of the car at an instant when its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{38ece0f6-1c29-4e7a-9d66-16c3e2b695f9-3_240_862_274_644} Particles \(P\) and \(Q\) start from points \(A\) and \(B\) respectively, at the same instant, and move towards each other in a horizontal straight line. The initial speeds of \(P\) and \(Q\) are \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The accelerations of \(P\) and \(Q\) are constant and equal to \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) respectively (see diagram).

1. Find the speed of \(P\) at the instant when the speed of \(P\) is 1.8 times the speed of \(Q\).
2. Given that \(A B = 51 \mathrm {~m}\), find the time taken from the start until \(P\) and \(Q\) meet.

1 A car travels in a straight line with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). It passes the points \(A , B\) and \(C\), in this order, with speeds \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 } , 7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The distances \(A B\) and \(B C\) are \(d _ { 1 } \mathrm {~m}\) and \(d _ { 2 } \mathrm {~m}\) respectively.

1. Write down an equation connecting
   1. \(d _ { 1 }\) and \(a\),
   2. \(d _ { 2 }\) and \(a\).
   3. Hence find \(d _ { 1 }\) in terms of \(d _ { 2 }\).

3 A car of mass 1200 kg is travelling on a horizontal straight road and passes through a point \(A\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power of the car's engine is 18 kW and the resistance to the car's motion is 900 N .

1. Find the deceleration of the car at \(A\).
2. Show that the speed of the car does not fall below \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while the car continues to move with the engine exerting a constant power of 18 kW .

5 \includegraphics[max width=\textwidth, alt={}, center]{a4cb105b-55d2-4793-95d2-3d791990a1f6-3_643_481_274_831} Particles \(A\) and \(B\), of masses 0.5 kg and \(m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(B\) is held at rest on the horizontal floor and particle \(A\) hangs in equilibrium (see diagram). \(B\) is released and each particle starts to move vertically. \(A\) hits the floor 2 s after \(B\) is released. The speed of each particle when \(A\) hits the floor is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. For the motion while \(A\) is moving downwards, find
   1. the acceleration of \(A\),
   2. the tension in the string.
   3. Find the value of \(m\).

6 A train travels from \(A\) to \(B\), a distance of 20000 m , taking 1000 s . The journey has three stages. In the first stage the train starts from rest at \(A\) and accelerates uniformly until its speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the second stage the train travels at constant speed \(V _ { \mathrm { m } } { } ^ { - 1 }\) for 600 s . During the third stage of the journey the train decelerates uniformly, coming to rest at \(B\).

1. Sketch the velocity-time graph for the train's journey.
2. Find the value of \(V\).
3. Given that the acceleration of the train during the first stage of the journey is \(0.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the distance travelled by the train during the third stage of the journey. \(7 \quad\) A particle \(P\) is held at rest at a fixed point \(O\) and then released. \(P\) falls freely under gravity until it reaches the point \(A\) which is 1.25 m below \(O\).
4. Find the speed of \(P\) at \(A\) and the time taken for \(P\) to reach \(A\). The particle continues to fall, but now its downward acceleration \(t\) seconds after passing through \(A\) is \(( 10 - 0.3 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
5. Find the total distance \(P\) has fallen, 3 s after being released from \(O\).

1 \includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-2_608_831_258_657} Two particles \(P\) and \(Q\) move vertically under gravity. The graphs show the upward velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particles at time \(t \mathrm {~s}\), for \(0 \leqslant t \leqslant 4 . P\) starts with velocity \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) starts from rest.

1. Find the value of \(V\). Given that \(Q\) reaches the horizontal ground when \(t = 4\), find
2. the speed with which \(Q\) reaches the ground,
3. the height of \(Q\) above the ground when \(t = 0\).

4 A particle \(P\) starts from a fixed point \(O\) at time \(t = 0\), where \(t\) is in seconds, and moves with constant acceleration in a straight line. The initial velocity of \(P\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its velocity when \(t = 10\) is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the displacement of \(P\) from \(O\) when \(t = 10\). Another particle \(Q\) also starts from \(O\) when \(t = 0\) and moves along the same straight line as \(P\). The acceleration of \(Q\) at time \(t\) is \(0.03 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that \(Q\) has the same velocity as \(P\) when \(t = 10\), show that it also has the same displacement from \(O\) as \(P\) when \(t = 10\).

5 Particles \(P\) and \(Q\) are projected vertically upwards, from different points on horizontal ground, with velocities of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. \(Q\) is projected 0.4 s later than \(P\). Find

1. the time for which \(P\) 's height above the ground is greater than 15 m ,
2. the velocities of \(P\) and \(Q\) at the instant when the particles are at the same height.

6 \includegraphics[max width=\textwidth, alt={}, center]{881993e1-71ea-4801-bfc8-40c17a1387a9-3_579_1518_258_315} The diagram shows the velocity-time graph for a particle \(P\) which travels on a straight line \(A B\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\). The graph consists of five straight line segments. The particle starts from rest when \(t = 0\) at a point \(X\) on the line between \(A\) and \(B\) and moves towards \(A\). The particle comes to rest at \(A\) when \(t = 2.5\).

1. Given that the distance \(X A\) is 4 m , find the greatest speed reached by \(P\) during this stage of the motion. In the second stage, \(P\) starts from rest at \(A\) when \(t = 2.5\) and moves towards \(B\). The distance \(A B\) is 48 m . The particle takes 12 s to travel from \(A\) to \(B\) and comes to rest at \(B\). For the first 2 s of this stage \(P\) accelerates at \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), reaching a velocity of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
2. the value of \(V\),
3. the value of \(t\) at which \(P\) starts to decelerate during this stage,
4. the deceleration of \(P\) immediately before it reaches \(B\). \(7 \quad\) A particle \(P\) travels in a straight line. It passes through the point \(O\) of the line with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t = 0\), where \(t\) is in seconds. \(P\) 's velocity after leaving \(O\) is given by $$\left( 0.002 t ^ { 3 } - 0.12 t ^ { 2 } + 1.8 t + 5 \right) \mathrm { m } \mathrm {~s} ^ { - 1 }$$ The velocity of \(P\) is increasing when \(0 < t < T _ { 1 }\) and when \(t > T _ { 2 }\), and the velocity of \(P\) is decreasing when \(T _ { 1 } < t < T _ { 2 }\).
5. Find the values of \(T _ { 1 }\) and \(T _ { 2 }\) and the distance \(O P\) when \(t = T _ { 2 }\).
6. Find the velocity of \(P\) when \(t = T _ { 2 }\) and sketch the velocity-time graph for the motion of \(P\).

1 A particle \(P\) is released from rest at a point on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. Find the speed of \(P\)

1. when it has travelled 0.9 m ,
2. 0.8 s after it is released.

4 \includegraphics[max width=\textwidth, alt={}, center]{28562a1b-ec9a-40d2-bbb3-729770688971-2_449_1273_1829_438} \(A , B\) and \(C\) are three points on a line of greatest slope of a smooth plane inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. \(A\) is higher than \(B\) and \(B\) is higher than \(C\), and the distances \(A B\) and \(B C\) are 1.76 m and 2.16 m respectively. A particle slides down the plane with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The speed of the particle at \(A\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). The particle takes 0.8 s to travel from \(A\) to \(B\) and takes 1.4 s to travel from \(A\) to \(C\). Find

1. the values of \(u\) and \(a\),
2. the value of \(\theta\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{28562a1b-ec9a-40d2-bbb3-729770688971-3_188_510_260_388} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{28562a1b-ec9a-40d2-bbb3-729770688971-3_196_570_255_1187} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A block of mass 2 kg is at rest on a horizontal floor. The coefficient of friction between the block and the floor is \(\mu\). A force of magnitude 12 N acts on the block at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). When the applied force acts downwards as in Fig. 1 the block remains at rest.

2 A block of mass 6 kg is sliding down a line of greatest slope of a plane inclined at \(8 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is 0.2 .

1. Find the deceleration of the block.
2. Given that the initial speed of the block is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find how far the block travels.

5 Particles \(A\) and \(B\), of masses 0.9 kg and 0.6 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. The system is released from rest with the string taut, with its straight parts vertical and with the particles at the same height above the horizontal floor. In the subsequent motion, \(B\) does not reach the pulley.

1. Find the acceleration of \(A\) and the tension in the string during the motion before \(A\) hits the floor. After \(A\) hits the floor, \(B\) continues to move vertically upwards for a further 0.3 s .
2. Find the height of the particles above the floor at the instant that they started to move.

1 \includegraphics[max width=\textwidth, alt={}, center]{155bc571-80e4-4c93-859f-bb150a109211-2_675_1380_255_379} A woman walks in a straight line. The woman's velocity \(t\) seconds after passing through a fixed point \(A\) on the line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The graph of \(v\) against \(t\) consists of 4 straight line segments (see diagram). The woman is at the point \(B\) when \(t = 60\). Find

1. the woman's acceleration for \(0 < t < 30\) and for \(30 < t < 40\),
2. the distance \(A B\),
3. the total distance walked by the woman.