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

A particle is moving in a straight line with constant acceleration \(a\) m s\(^{-2}\) The particle's velocity, \(v\) m s\(^{-1}\), varies with time, \(t\) seconds, so that $$v = 3 - 4t$$ Deduce the value of \(a\) Circle your answer. [1 mark] \(-4\) \qquad \(-1\) \qquad \(3\) \qquad \(4\)

A graph indicating how the velocity, \(v\) m s\(^{-1}\), of a particle changes with respect to time, \(t\) seconds, is shown below. \includegraphics{figure_15}

1. Find the total distance travelled by the particle over the 8 second period shown. [3 marks]
2. A student claims that "The displacement of the particle is less than the distance travelled." State the range of values of \(t\) for which this claim is true. [1 mark]

A particle moves in a straight line with acceleration \(a\) m s\(^{-2}\), at time \(t\) seconds, where $$a = 10 - 6t$$ The particle's velocity, \(v\) m s\(^{-1}\), and displacement, \(r\) metres, are both initially zero. Show that $$r = t^2(5 - t)$$ Fully justify your answer. [4 marks]

At time \(t\) seconds a particle, \(P\), has position vector \(\mathbf{r}\) metres, with respect to a fixed origin, such that $$\mathbf{r} = (t^3 - 5t^2)\mathbf{i} + (8t - t^2)\mathbf{j}$$

1. Find the exact speed of \(P\) when \(t = 2\) [4 marks]
2. Bella claims that the magnitude of acceleration of \(P\) will never be zero. Determine whether Bella's claim is correct. Fully justify your answer. [3 marks]

The displacement, \(r\) metres, of a particle at time \(t\) seconds is $$r = 6t - 2t^2$$

1. Find the value of \(r\) when \(t = 4\) [1 mark]
2. Determine the range of values of \(t\) for which the displacement is positive. [2 marks]

A particle is moving in a straight line through the origin \(O\) The displacement of the particle, \(r\) metres, from \(O\), at time \(t\) seconds is given by $$r = p + 2t - qe^{-0.2t}$$ where \(p\) and \(q\) are constants. When \(t = 3\), the acceleration of the particle is \(-1.8\) m s\(^{-2}\)

1. Show that \(q \approx 82\) [5 marks]
2. The particle has an initial displacement of 5 metres. Find the value of \(p\) Give your answer to two significant figures. [2 marks]

A turntable rotates at a constant speed of \(33\frac{1}{3}\) revolutions per minute. Find the angular speed in radians per second. Circle your answer. [1 mark] \(\frac{5\pi}{9}\) \quad \(\frac{10\pi}{9}\) \quad \(\frac{5\pi}{3}\) \quad \(\frac{20\pi}{9}\)

A particle P moves so that its position vector \(\mathbf{r}\) at time \(t\) is given by $$\mathbf{r} = (5 + 20t)\mathbf{i} + (95 + 10t - 5t^2)\mathbf{j}.$$

1. Determine the initial velocity of P. [3] At time \(t = T\), P is moving in a direction perpendicular to its initial direction of motion.
2. Determine the value of \(T\). [3]
3. Determine the distance of P from its initial position at time \(T\). [4]

A particle \(P\), of mass 3 kg, moves along the horizontal \(x\)-axis under the action of a resultant force \(F\) N. Its velocity \(v\) ms\(^{-1}\) at time \(t\) seconds is given by $$v = 12t - 3t^2.$$

1. Given that the particle is at the origin \(O\) when \(t = 1\), find an expression for the displacement of the particle from \(O\) at time \(t\) s. [3]
2. Find an expression for the acceleration of the particle at time \(t\) s. [2]

Two particles \(A\) and \(B\) have position vectors \(\mathbf{r}_A\) metres and \(\mathbf{r}_B\) metres at time \(t\) seconds, where $$\mathbf{r}_A = t^2\mathbf{i} + (3t - 1)\mathbf{j} \quad \text{and} \quad \mathbf{r}_B = (1 - 2t^2)\mathbf{i} + (3t - 2t^2)\mathbf{j}, \quad \text{for } t \geqslant 0.$$

1. Find the values of \(t\) when \(A\) and \(B\) are moving with the same speed. [5]
2. Show that the distance, \(d\) metres, between \(A\) and \(B\) at time \(t\) satisfies $$d^2 = 13t^4 - 10t^2 + 2.$$ [3]
3. Hence find the shortest distance between \(A\) and \(B\) in the subsequent motion. [6]

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 travels along a straight line. Its velocity \(v\) m s\(^{-1}\) after \(t\) seconds is given by $$v = t^3 - 9t^2 + 20t$$ When \(t = 0\), the particle is at rest at \(P\).

1. Find the times, other than \(t = 0\), at which the particle is at rest. [2]
2. Find the displacement of the particle from \(P\) when \(t = 2\). [4]

A particle travels along a straight line. Its velocity \(v\) ms\(^{-1}\) after \(t\) seconds is given by $$v = t^3 - 9t^2 + 20t$$ When \(t = 0\), the particle is at rest at \(P\).

1. Find the times, other than \(t = 0\), at which the particle is at rest. [2]
2. Find the displacement of the particle from \(P\) when \(t = 2\). [4]