Displacement from velocity by integration

5 A particle \(P\) moves in a straight line that passes through the origin \(O\). The velocity of \(P\) at time \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 20 t - t ^ { 3 }\). At time \(t = 0\) the particle is at rest at a point whose displacement from \(O\) is - 36 m .

1. Find an expression for the displacement of \(P\) from \(O\) in terms of \(t\).
2. Find the displacement of \(P\) from \(O\) when \(t = 4\).
3. Find the values of \(t\) for which the particle is at \(O\).

5 A particle \(P\) moves along the \(x\)-axis in the positive direction. The velocity of \(P\) at time \(t \mathrm {~s}\) is \(0.03 t ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 5\) the displacement of \(P\) from the origin \(O\) is 2.5 m .

1. Find an expression, in terms of \(t\), for the displacement of \(P\) from \(O\).
2. Find the velocity of \(P\) when its displacement from \(O\) is 11.25 m .

2 A motorcyclist starts from rest at \(A\) and travels in a straight line until he comes to rest again at \(B\). The velocity of the motorcyclist \(t\) seconds after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = t - 0.01 t ^ { 2 }\). Find

1. the time taken for the motorcyclist to travel from \(A\) to \(B\),
2. the distance \(A B\).

2 A particle starts at a point \(O\) and moves along a straight line. Its velocity \(t\) s after leaving \(O\) is \(\left( 1.2 t - 0.12 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the displacement of the particle from \(O\) when its acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

4 A particle \(P\) moves in a straight line starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is given by \(v = ( 2 t - 5 ) ^ { 3 }\).

1. Find the values of \(t\) when the acceleration of \(P\) is \(54 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find an expression for the displacement of \(P\) from \(O\) at time \(t \mathrm {~s}\).

4 The velocity of a particle \(t \mathrm {~s}\) after it starts from rest is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 1.25 t - 0.05 t ^ { 2 }\). Find

1. the initial acceleration of the particle,
2. the displacement of the particle from its starting point at the instant when its acceleration is \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

3 A particle \(P\) moves in a straight line. It starts from a point \(O\) on the line with velocity \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(0.8 t ^ { - 0.75 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the displacement of \(P\) from \(O\) when \(t = 16\).

2 A particle moves in a straight line. Its velocity \(t\) seconds after leaving a fixed point \(O\) on the line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.2 t + 0.006 t ^ { 2 }\). For the instant when the acceleration of the particle is 2.5 times its initial acceleration,

1. show that \(t = 25\),
2. find the displacement of the particle from \(O\).

5 A particle \(P\) starts from a fixed point \(O\) and moves in a straight line. At time \(t\) s after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is given by \(v = 6 t - 0.3 t ^ { 2 }\). The particle comes to instantaneous rest at point \(X\).

1. Find the distance \(O X\). A second particle \(Q\) starts from rest from \(O\), at the same instant as \(P\), and also travels in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of \(Q\) is given by \(a = k - 12 t\), where \(k\) is a constant. The displacement of \(Q\) from \(O\) is 400 m when \(t = 10\).
2. Find the value of \(k\).

4 A particle moves in a straight line. Its velocity \(t \mathrm {~s}\) after leaving a fixed point on the line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = t + 0.1 t ^ { 2 }\). Find

1. an expression for the acceleration of the particle at time \(t\),
2. the distance travelled by the particle from time \(t = 0\) until the instant when its acceleration is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

6 The speed of a 100 metre runner in \(\mathrm { ms } ^ { - 1 }\) is measured electronically every 4 seconds.
The measurements are plotted as points on the speed-time graph in Fig. 6. The vertical dotted line is drawn through the runner's finishing time. Fig. 6 also illustrates Model P in which the points are joined by straight lines. \begin{figure}[h] \end{figure}

1. Use Model P to estimate
   (A) the distance the runner has gone at the end of 12 seconds,
   (B) how long the runner took to complete 100 m . A mathematician proposes Model Q in which the runner's speed, \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\), is given by $$v = \frac { 5 } { 2 } t - \frac { 1 } { 8 } t ^ { 2 }$$
2. Verify that Model Q gives the correct speed for \(t = 8\).
3. Use Model Q to estimate the distance the runner has gone at the end of 12 seconds.
4. The runner was timed at 11.35 seconds for the 100 m . Which model places the runner closer to the finishing line at this time?
5. Find the greatest acceleration of the runner according to each model.

6 The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of a particle at time \(t \mathrm {~s}\) is given by \(v = t ^ { 2 } - 9\). The particle travels in a straight line and passes through a fixed point \(O\) when \(t = 2\).

1. Find the displacement of the particle from \(O\) when \(t = 0\).
2. Calculate the distance the particle travels from its position at \(t = 0\) until it changes its direction of motion.
3. Calculate the distance of the particle from \(O\) when the acceleration of the particle is \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

3 A particle \(P\) travels in a straight line. The velocity of \(P\) at time \(t\) seconds after it passes through a fixed point \(A\) is given by \(\left( 0.6 t ^ { 2 } + 3 \right) \mathrm { ms } ^ { - 1 }\). Find

1. the velocity of \(P\) when it passes through \(A\),
2. the displacement of \(P\) from \(A\) when \(t = 1.5\),
3. the velocity of \(P\) when it has acceleration \(6 \mathrm {~ms} ^ { - 2 }\).

5 The velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of a particle moving along a straight line is given by $$v = 3 t ^ { 2 } - 12 t + 14$$ where \(t\) is the time in seconds.

1. Find an expression for the acceleration of the particle at time \(t\).
2. Find the displacement of the particle from its position when \(t = 1\) to its position when \(t = 3\).
3. You are given that \(v\) is always positive. Explain how this tells you that the distance travelled by the particle between \(t = 1\) and \(t = 3\) has the same value as the displacement between these times.
   [0pt] [2]

1 A curve passes through the point \(( 0,1 )\) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { 1 + x ^ { 2 } }$$ Starting at the point \(( 0,1 )\), use a step-by-step method with a step length of 0.2 to estimate the value of \(y\) at \(x = 0.4\). Give your answer to five decimal places.

1 A curve passes through the point (1,3) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x } { 1 + x ^ { 3 } }$$ Starting at the point ( 1,3 ), use a step-by-step method with a step length of 0.1 to estimate the value of \(y\) at \(x = 1.2\). Give your answer to four decimal places.

1 A curve passes through the point ( 1,3 ) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 + x ^ { 3 }$$ Starting at the point ( 1,3 ), use a step-by-step method with a step length of 0.1 to estimate the \(y\)-coordinate of the point on the curve for which \(x = 1.3\). Give your answer to three decimal places.
(No credit will be given for methods involving integration.)

1 A curve passes through the point \(( 2,3 )\) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 2 + x } }$$ Starting at the point \(( 2,3 )\), use a step-by-step method with a step length of 0.5 to estimate the value of \(y\) at \(x = 3\). Give your answer to four decimal places.

1 A curve passes through the point \(( 9,6 )\) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 + \sqrt { x } }$$ Use a step-by-step method with a step length of 0.25 to estimate the value of \(y\) at \(x = 9.5\). Give your answer to four decimal places.
[0pt] [5 marks]

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where

$$v = 5 \sin 2 t$$ When \(t = 0 , x = 1\) and \(P\) is at rest.

1. Find the magnitude and direction of the acceleration of \(P\) at the instant when \(P\) is next at rest.
2. Show that \(1 \leqslant x \leqslant 6\)
3. Find the total time, in the first \(4 \pi\) seconds of the motion, for which \(P\) is more than 3 metres from \(O\)
   \includegraphics[max width=\textwidth, alt={}]{a7901165-1679-4d30-9444-0c27020e32ea-16_2260_52_309_1982}

1 A particle moves along a straight line. At time \(t\), it has velocity \(v\), where $$v = 4 t ^ { 3 } - 8 \sin 2 t + 5$$ When \(t = 0\), the particle is at the origin.
Find an expression for the displacement of the particle from the origin at time \(t\).

16 A remote-controlled toy car is moving over a horizontal surface. It moves in a straight line through a point \(A\). The toy is initially at the point with displacement 3 metres from \(A\). Its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds is defined by $$v = 0.06 \left( 2 + t - t ^ { 2 } \right)$$ 16

1. Find an expression for the displacement, \(r\) metres, of the toy from \(A\) at time \(t\) seconds.
   16
2. In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) At time \(t = 2\) seconds, the toy launches a ball which travels directly upwards with initial speed \(3.43 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the time taken for the ball to reach its highest point.
   \includegraphics[max width=\textwidth, alt={}, center]{c982106c-b742-444f-aeed-6f59ff3fae56-24_2496_1721_214_150}

13 A car, starting from rest, is driven along a horizontal track. The velocity of the car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds, is modelled by the equation $$v = 0.48 t ^ { 2 } - 0.024 t ^ { 3 } \text { for } 0 \leq t \leq 15$$ 13

1. Find the distance the car travels during the first 10 seconds of its journey.
   13
2. Find the maximum speed of the car.
   Give your answer to three significant figures.
   13
3. Deduce the range of values of \(t\) for which the car is modelled as decelerating.

15 A driver is road-testing two minibuses, \(A\) and \(B\), for a taxi company. The performance of each minibus along a straight track is compared.
A flag is dropped to indicate the start of the test.
Each minibus starts from rest.
The acceleration in \(\mathrm { ms } ^ { - 2 }\) of each minibus is modelled as a function of time, \(t\) seconds, after the flag is dropped: The acceleration of \(\mathrm { A } = 0.138 t ^ { 2 }\)
The acceleration of \(\mathrm { B } = 0.024 t ^ { 3 }\)
15

1. Find the time taken for A to travel 100 metres.
   Give your answer to four significant figures.
   15
2. The company decides to buy the minibus which travels 100 metres in the shortest time. Determine which minibus should be bought.
   15
3. The models assume that both minibuses start moving immediately when \(t = 0\) In light of this, explain why the company may, in reality, make the wrong decision.
   A particle is projected with an initial speed \(u\), at an angle of \(35 ^ { \circ }\) above the horizontal.
   It lands at a point 10 metres vertically below its starting position.
   The particle takes 1.5 seconds to reach the highest point of its trajectory.