Variable acceleration with initial conditions

A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds is \((4t - 8)\) m s\(^{-2}\), measured in the direction of \(x\) increasing. The velocity of \(P\) at time \(t\) seconds is \(v\) m s\(^{-1}\). Given that \(v = 6\) when \(t = 0\), find

1. \(v\) in terms of \(t\), [4]

1. the distance between the two points where \(P\) is instantaneously at rest. [7]

A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds is \((4t - 8)\) m s\(^{-2}\), measured in the direction of \(x\) increasing. The velocity of \(P\) at time \(t\) seconds is \(v\) m s\(^{-1}\). Given that \(v = 6\) when \(t = 0\), find

1. \(v\) in terms of \(t\), [4]
2. the distance between the two points where \(P\) is instantaneously at rest. [7]

A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds is \((t - 4)\) m s\(^{-2}\) in the positive \(x\)-direction. The velocity of \(P\) at time \(t\) seconds is \(v\) m s\(^{-1}\). When \(t = 0\), \(v = 6\). Find

1. \(v\) in terms of \(t\), [4]
2. the values of \(t\) when \(P\) is instantaneously at rest, [3]
3. the distance between the two points at which \(P\) is instantaneously at rest. [4]

A particle \(P\) moves in a straight line so that, at time \(t\) seconds after leaving a fixed point \(O\), its acceleration is \(-\frac{1}{10}t \text{ m s}^{-2}\). At time \(t = 0\), the velocity of \(P\) is \(V \text{ m s}^{-1}\).

1. Find, by integration, an expression in terms of \(t\) and \(V\) for the velocity of \(P\). [4]
2. Find the value of \(V\), given that \(P\) is instantaneously at rest when \(t = 10\). [2]
3. Find the displacement of \(P\) from \(O\) when \(t = 10\). [4]
4. Find the speed with which the particle returns to \(O\). [3]

A particle, \(P\), is moving in a straight line with acceleration \(a\text{ m s}^{-2}\) at time \(t\) seconds, where $$a = 4 - 3t^2$$

1. Initially \(P\) is stationary. Find an expression for the velocity of \(P\) in terms of \(t\). [2 marks]
2. When \(t = 2\), the displacement of \(P\) from a fixed point, O, is 39 metres. Find the time at which \(P\) passes through O, giving your answer to three significant figures. Fully justify your answer. [5 marks]

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 m s\(^{-2}\) of each minibus is modelled as a function of time, \(t\) seconds, after the flag is dropped: The acceleration of A = \(0.138 t^2\) The acceleration of B = \(0.024 t^3\)

1. Find the time taken for A to travel 100 metres. Give your answer to four significant figures. [4 marks]
2. The company decides to buy the minibus which travels 100 metres in the shortest time. Determine which minibus should be bought. [4 marks]
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. [1 mark]

A particle \(P\) is free to move along a straight line \(Ox\). It starts from rest at \(O\) and after \(t\) seconds its acceleration \(a\,\mathrm{m}\,\mathrm{s}^{-2}\) is given by \(a = 12 - 6t\).

1. Find an expression in terms of \(t\) for its velocity \(v\,\mathrm{m}\,\mathrm{s}^{-1}\). Hence find the velocity of \(P\) when \(t = 4\). [4]
2. Find the displacement of \(P\) from \(O\) when \(t = 4\). [3]
3. Find the velocity of \(P\) when it returns to \(O\). [3]

A particle \(P\) is free to move along a straight line \(Ox\). It starts from rest at \(O\) and after \(t\) seconds its acceleration \(a \mathrm{~m} \mathrm{~s}^{-2}\) is given by \(a = 12 - 6t\).

1. Find an expression in terms of \(t\) for its velocity \(v \mathrm{~m} \mathrm{~s}^{-1}\). Hence find the velocity of \(P\) when \(t = 4\). [4]
2. Find the displacement of \(P\) from \(O\) when \(t = 4\). [3]
3. Find the velocity of \(P\) when it returns to \(O\). [3]

A particle \(P\) is free to move along a straight line \(Ox\). It starts from rest at \(O\) and after \(t\) seconds its acceleration \(a \mathrm{m} \mathrm{s}^{-2}\) is given by \(a = 12 - 6t\).

1. Find an expression in terms of \(t\) for its velocity \(v \mathrm{m} \mathrm{s}^{-1}\). Hence find the velocity of \(P\) when \(t = 4\). [4]
2. Find the displacement of \(P\) from \(O\) when \(t = 4\). [3]
3. Find the velocity of \(P\) when it returns to \(O\). [3]