3.02a Kinematics language: position, displacement, velocity, acceleration

A particle \(P\) moves in a straight line starting from a point \(O\) and comes to rest \(35\) s later. At time \(t\) s after leaving \(O\), the velocity \(v\) m s\(^{-1}\) of \(P\) is given by $$v = \frac{4}{5}t^2 \quad 0 \leq t \leq 5,$$ $$v = 2t + 10 \quad 5 \leq t \leq 15,$$ $$v = a + bt^2 \quad 15 \leq t \leq 35,$$ where \(a\) and \(b\) are constants such that \(a > 0\) and \(b < 0\).

1. Show that the values of \(a\) and \(b\) are \(49\) and \(-0.04\) respectively. [3]
2. Sketch the velocity-time graph. [4]
3. Find the total distance travelled by \(P\) during the \(35\) s. [5]

A particle moves in a straight line. The displacement of the particle at time \(t\) s is \(s\) m, where $$s = t^3 - 6t^2 + 4t.$$ Find the velocity of the particle at the instant when its acceleration is zero. [4]

A particle \(P\) moves in a straight line, starting from a point \(O\). The velocity of \(P\), measured in m s\(^{-1}\), at time \(t\) s after leaving \(O\) is given by $$v = 0.6t - 0.03t^2.$$

1. Verify that, when \(t = 5\), the particle is 6.25 m from \(O\). Find the acceleration of the particle at this time. [4]
2. Find the values of \(t\) at which the particle is travelling at half of its maximum velocity. [6]

A particle moves in a straight line. Its displacement from a fixed point \(O\) at time \(t\) seconds is \(s\) metres, where \(s = t^3 - 9t^2 + 24t\).

1. Find expressions for the velocity \(v\) and acceleration \(a\) of the particle at time \(t\).
2. Find the values of \(t\) for which the particle is at rest.
3. Find the total distance travelled by the particle in the first \(6\) seconds.

[8]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors directed due east and due north respectively.] A particle \(P\) is moving with constant velocity \((-6\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\). At time \(t = 0\), \(P\) passes through the point with position vector \((21\mathbf{i} + 5\mathbf{j})\) m, relative to a fixed origin \(O\).

1. Find the direction of motion of \(P\), giving your answer as a bearing to the nearest degree. [3]
2. Write down the position vector of \(P\) at time \(t\) seconds. [1]
3. Find the time at which \(P\) is north-west of \(O\). [3]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin.] At 2 pm, the position vector of ship \(P\) is \((5\mathbf{i} - 3\mathbf{j})\) km and the position vector of ship \(Q\) is \((7\mathbf{i} + 5\mathbf{j})\) km.

1. Find the distance between \(P\) and \(Q\) at 2 pm. [3]

Ship \(P\) is moving with constant velocity \((2\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\) and ship \(Q\) is moving with constant velocity \((-3\mathbf{i} - 15\mathbf{j})\) km h\(^{-1}\).

1. Find the position vector of \(P\) at time \(t\) hours after 2 pm. [2]
2. Find the position vector of \(Q\) at time \(t\) hours after 2 pm. [1]
3. Show that \(Q\) will meet \(P\) and find the time at which they meet. [5]
4. Find the position vector of the point at which they meet. [2]

[In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).] Two ships, \(A\) and \(B\), are moving with constant velocities. The velocity of \(A\) is \((3\mathbf{i} + 12\mathbf{j})\text{ kmh}^{-1}\) and the velocity of \(B\) is \((p\mathbf{i} + q\mathbf{j})\text{ kmh}^{-1}\)

1. Find the speed of \(A\). [2] The ships are modelled as particles. At 12 noon, \(A\) is at the point with position vector \((-9\mathbf{i} + 6\mathbf{j})\) km and \(B\) is at the point with position vector \((16\mathbf{i} + 6\mathbf{j})\) km. At time \(t\) hours after 12 noon, $$\overrightarrow{AB} = [(25 - 12t)\mathbf{i} - 9t\mathbf{j}] \text{ km}$$
2. Find the value of \(p\) and the value of \(q\). [7]
3. Find the bearing of \(A\) from \(B\) when the ships are 15 km apart, giving your answer to the nearest degree. [7]

A particle \(P\) is moving with constant velocity \((-3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\). At time \(t = 6\) s \(P\) is at the point with position vector \((-4\mathbf{i} - 7\mathbf{j})\) m. Find the distance of \(P\) from the origin at time \(t = 2\) s. [5]

Two cars \(P\) and \(Q\) are moving in the same direction along the same straight horizontal road. Car \(P\) is moving with constant speed 25 m s\(^{-1}\). At time \(t = 0\), \(P\) overtakes \(Q\) which is moving with constant speed 20 m s\(^{-1}\). From \(t = 7\) seconds, \(P\) decelerates uniformly, coming to rest at a point \(X\) which is 800 m from the point where \(P\) overtook \(Q\). From \(t = 25\) s, \(Q\) decelerates uniformly, coming to rest at the same point \(X\) at the same instant as \(P\).

1. Sketch, on the same axes, the speed-time graphs of the two cars for the period from \(t = 0\) to the time when they both come to rest at the point \(X\). [4]
2. Find the value of \(T\). [8]

A motor scooter and a van set off along a straight road. They both start from rest at the same time and level with each other. The scooter accelerates with constant acceleration until it reaches its top speed of 20 m s\(^{-1}\). It then maintains a constant speed of 20 m s\(^{-1}\). The van accelerates with constant acceleration for 10 s until it reaches its top speed \(V\) m s\(^{-1}\), \(V > 20\). It then maintains a constant speed of \(V\) m s\(^{-1}\). The van draws level with the scooter when the scooter has been travelling for 40 s at its top speed. The total distance travelled by each vehicle is then 850 m.

1. Sketch on the same diagram the speed-time graphs of both vehicles to illustrate their motion from the time when they start to the time when the van overtakes the scooter. [3]
2. Find the time for which the scooter is accelerating. [3]
3. Find the top speed of the van. [3]

[In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship \(S\) is moving along a straight line with constant velocity. At time \(t\) hours the position vector of \(S\) is \(\mathbf{s}\) km. When \(t = 0\), \(\mathbf{s} = 9\mathbf{i} - 6\mathbf{j}\). When \(t = 4\), \(\mathbf{s} = 21\mathbf{i} + 10\mathbf{j}\). Find

1. the speed of \(S\), [4]
2. the direction in which \(S\) is moving, giving your answer as a bearing. [2]
3. Show that \(\mathbf{s} = (3t + 9)\mathbf{i} + (4t - 6)\mathbf{j}\). [2]

A lighthouse \(L\) is located at the point with position vector \((18\mathbf{i} + 6\mathbf{j})\) km. When \(t = T\), the ship \(S\) is 10 km from \(L\).

1. Find the possible values of \(T\). [6]

[In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship sets sail at 9 am from a port \(P\) and moves with constant velocity. The position vector of \(P\) is \((4\mathbf{i} - 8\mathbf{j})\) km. At 9.30 am the ship is at the point with position vector \((\mathbf{i} - 4\mathbf{j})\) km.

1. Find the speed of the ship in km h\(^{-1}\). [4]
2. Show that the position vector \(\mathbf{r}\) km of the ship, \(t\) hours after 9 am, is given by \(\mathbf{r} = (4 - 6t)\mathbf{i} + (8t - 8)\mathbf{j}\). [2]

At 10 am, a passenger on the ship observes that a lighthouse \(L\) is due west of the ship. At 10.30 am, the passenger observes that \(L\) is now south-west of the ship.

1. Find the position vector of \(L\). [5]

A particle \(P\) of mass \(2 \text{ kg}\) moves in a plane under the action of a single constant force \(\mathbf{F}\) newtons. At time \(t\) seconds, the velocity of \(P\) is \(\mathbf{v} \text{ m s}^{-1}\). When \(t = 0\), \(\mathbf{v} = (-5\mathbf{i} + 7\mathbf{j})\) and when \(t = 3\), \(\mathbf{v} = (\mathbf{i} - 2\mathbf{j})\).

1. Find in degrees the angle between the direction of motion of \(P\) when \(t = 3\) and the vector \(\mathbf{j}\). [3]
2. Find the acceleration of \(P\). [2]
3. Find the magnitude of \(\mathbf{F}\). [3]
4. Find in terms of \(t\) the velocity of \(P\). [2]
5. Find the time at which \(P\) is moving parallel to the vector \(\mathbf{i} + \mathbf{j}\). [3]

A small boat \(S\), drifting in the sea, is modelled as a particle moving in a straight line at constant speed. When first sighted at 0900, \(S\) is at a point with position vector \((4\mathbf{i} - 6\mathbf{j})\) km relative to a fixed origin \(O\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively. At 0945, \(S\) is at the point with position vector \((7\mathbf{i} - 7.5\mathbf{j})\) km. At time \(t\) hours after 0900, \(S\) is at the point with position vector \(\mathbf{s}\) km.

1. Calculate the bearing on which \(S\) is drifting. [4]
2. Find an expression for \(\mathbf{s}\) in terms of \(t\). [3]

At 1000 a motor boat \(M\) leaves \(O\) and travels with constant velocity \((p\mathbf{i} + q\mathbf{j})\) km h\(^{-1}\). Given that \(M\) intercepts \(S\) at 1015,

1. calculate the value of \(p\) and the value of \(q\). [6]

In taking off, an aircraft moves on a straight runway \(AB\) of length 1.2 km. The aircraft moves from \(A\) with initial speed \(2 \text{ m s}^{-1}\). It moves with constant acceleration and 20 s later it leaves the runway at \(C\) with speed \(74 \text{ m s}^{-1}\). Find

1. the acceleration of the aircraft, [2]
2. the distance \(BC\). [4]

A train is travelling at \(10 \text{ m s}^{-1}\) on a straight horizontal track. The driver sees a red signal 135 m ahead and immediately applies the brakes. The train immediately decelerates with constant deceleration for 12 s, reducing its speed to \(3 \text{ m s}^{-1}\). The driver then releases the brakes and allows the train to travel at a constant speed of \(3 \text{ m s}^{-1}\) for a further 15 s. He then applies the brakes again and the train slows down with constant deceleration, coming to rest as it reaches the signal.

1. Sketch a speed-time graph to show the motion of the train, [3]
2. Find the distance travelled by the train from the moment when the brakes are first applied to the moment when its speed first reaches \(3 \text{ m s}^{-1}\). [2]
3. Find the total time from the moment when the brakes are first applied to the moment when the train comes to rest. [5]

[In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal vectors due east and north respectively.] At time \(t = 0\), a football player kicks a ball from the point \(A\) with position vector \((2\mathbf{i} + \mathbf{j})\) m on a horizontal football field. The motion of the ball is modelled as that of a particle moving horizontally with constant velocity \((5\mathbf{i} + 8\mathbf{j}) \text{ m s}^{-1}\). Find

1. the speed of the ball, [2]
2. the position vector of the ball after \(t\) seconds. [2]

The point \(B\) on the field has position vector \((10\mathbf{i} + 7\mathbf{j})\) m.

1. Find the time when the ball is due north of \(B\). [2]

At time \(t = 0\), another player starts running due north from \(B\) and moves with constant speed \(v \text{ m s}^{-1}\). Given that he intercepts the ball,

1. find the value of \(v\). [6]
2. State one physical factor, other than air resistance, which would be needed in a refinement of the model of the ball's motion to make the model more realistic. [1]

Three posts \(P\), \(Q\) and \(R\) are fixed in that order at the side of a straight horizontal road. The distance from \(P\) to \(Q\) is 45 m and the distance from \(Q\) to \(R\) is 120 m. A car is moving along the road with constant acceleration \(a\) m s\(^{-2}\). The speed of the car, as it passes \(P\), is \(u\) m s\(^{-1}\). The car passes \(Q\) two seconds after passing \(P\), and the car passes \(R\) four seconds after passing \(Q\). Find

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

[7]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively.] A hiker \(H\) is walking with constant velocity \((1.2\mathbf{i} - 0.9\mathbf{j})\) m s\(^{-1}\).

1. Find the speed of \(H\). [2]

\includegraphics{figure_3} A horizontal field \(OABC\) is rectangular with \(OA\) due east and \(OC\) due north, as shown in Figure 3. At twelve noon hiker \(H\) is at the point \(Y\) with position vector \(100\mathbf{j}\) m, relative to the fixed origin \(O\).

1. Write down the position vector of \(H\) at time \(t\) seconds after noon. [2]

At noon, another hiker \(K\) is at the point with position vector \((9\mathbf{i} + 46\mathbf{j})\) m. Hiker \(K\) is moving with constant velocity \((0.75\mathbf{i} + 1.8\mathbf{j})\) m s\(^{-1}\).

1. Show that, at time \(t\) seconds after noon, $$\overrightarrow{HK} = [(9 - 0.45t)\mathbf{i} + (2.7t - 54)\mathbf{j}] \text{ metres.}$$ [4]

Hence,

1. show that the two hikers meet and find the position vector of the point where they meet. [5]

Two cars \(P\) and \(Q\) are moving in the same direction along the same straight horizontal road. Car \(P\) is moving with constant speed 25 m s\(^{-1}\). At time \(t = 0\), \(P\) overtakes \(Q\) which is moving with constant speed 20 m s\(^{-1}\). From \(t = T\) seconds, \(P\) decelerates uniformly, coming to rest at a point \(X\) which is 800 m from the point where \(P\) overtook \(Q\). From \(t = 25\) s, \(Q\) decelerates uniformly, coming to rest at the same point \(X\) at the same instant as \(P\).

1. Sketch, on the same axes, the speed-time graphs of the two cars for the period from \(t = 0\) to the time when they both come to rest at the point \(X\). [4]
2. Find the value of \(T\). [8]

Two cars \(A\) and \(B\) are moving on straight horizontal roads with constant velocities. The velocity of \(A\) is \(20 \text{ m s}^{-1}\) due east, and the velocity of \(B\) is \((10\mathbf{i} + 10\mathbf{j}) \text{ m s}^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors directed due east and due north respectively. Initially \(A\) is at the fixed origin \(O\), and the position vector of \(B\) is \(300\mathbf{j}\) m relative to \(O\). At time \(t\) seconds, the position vectors of \(A\) and \(B\) are \(\mathbf{r}\) metres and \(\mathbf{s}\) metres respectively.

1. Find expressions for \(\mathbf{r}\) and \(\mathbf{s}\) in terms of \(t\). [3]
2. Hence write down an expression for \(\overrightarrow{AB}\) in terms of \(t\). [1]
3. Find the time when the bearing of \(B\) from \(A\) is \(045°\). [5]
4. Find the time when the cars are again 300 m apart. [6]

At time \(t\) seconds, a particle \(P\) has position vector \(r\) metres relative to a fixed origin \(O\), where $$r = (t^2 + 2t)\mathbf{i} + (t - 2t^2)\mathbf{j}.$$ Show that the acceleration of \(P\) is constant and find its magnitude. [5]

A van of mass 1500 kg is driving up a straight road inclined at an angle \(α\) to the horizontal, where \(\sin α = \frac{1}{16}\). The resistance to motion due to non-gravitational forces is modelled as a constant force of magnitude 1000 N. Given that initially the speed of the van is 30 m s\(^{-1}\) and that the van's engine is operating at a rate of 60 kW,

1. calculate the magnitude of the initial deceleration of the van. [4]

When travelling up the same hill, the rate of working of the van's engine is increased to 80 kW. Using the same model for the resistance due to non-gravitational forces,

1. calculate in m s\(^{-1}\) the constant speed which can be sustained by the van at this rate of working. [4]

1. Give one reason why the use of this model for resistance may mean that your answer to part (b) is too high. [1]

A particle \(P\) of mass \(0.3\) kg is moving under the action of a single force \(F\) newtons. At time \(t\) seconds the velocity of \(P\), v m s\(^{-1}\), is given by $$\mathbf{v} = 3t^2\mathbf{i} + (6t - 4)\mathbf{j}.$$

1. Calculate, to 3 significant figures, the magnitude of \(\mathbf{F}\) when \(t = 2\). [5]

When \(t = 0\), \(P\) is at the point \(A\). The position vector of \(A\) with respect to a fixed origin \(O\) is \((3\mathbf{i} - 4\mathbf{j})\) m. When \(t = 4\), \(P\) is at the point \(B\).

1. Find the position vector of \(B\). [5]

The velocity v m s\(^{-1}\) of a particle \(P\) at time \(t\) seconds is given by $$\mathbf{v} = (3t - 2)\mathbf{i} - 5t\mathbf{j}.$$

1. Show that the acceleration of \(P\) is constant. [2]

At \(t = 0\), the position vector of \(P\) relative to a fixed origin \(O\) is 3i m.

1. Find the distance of \(P\) from \(O\) when \(t = 2\). [6]