3.02f Non-uniform acceleration: using differentiation and integration

A particle moves in a straight line starting from a point \(O\). The velocity \(v\) m s\(^{-1}\) of the particle \(t\) s after leaving \(O\) is given by $$v = t^3 - \frac{9}{2}t^2 + 1 \text{ for } 0 \leqslant t \leqslant 4.$$ You may assume that the velocity of the particle is positive for \(t < \frac{1}{2}\), is zero at \(t = \frac{1}{2}\) and is negative for \(t > \frac{1}{2}\).

1. Find the distance travelled between \(t = 0\) and \(t = \frac{1}{2}\). [4]
2. Find the positive value of \(t\) at which the acceleration is zero. Hence find the total distance travelled between \(t = 0\) and this instant. [4]

A particle \(P\) moves in a straight line. It starts from rest at a point \(O\) on the line and at time \(t\) s after leaving \(O\) it has acceleration \(a \text{ m s}^{-2}\), where \(a = 6t - 18\). Find the distance \(P\) moves before it comes to instantaneous rest. [6]

A particle \(P\) moves on the \(x\)-axis from the origin \(O\) with an initial velocity of \(-20\) m s\(^{-1}\). The acceleration \(a\) m s\(^{-2}\) at time \(t\) s after leaving \(O\) is given by \(a = 12 - 2t\).

1. Sketch a velocity-time graph for \(0 \leq t \leq 12\), indicating the times when \(P\) is at rest. [5]
2. Find the total distance travelled by \(P\) in the interval \(0 \leq t \leq 12\). [5]

A particle \(P\) travels in a straight line, starting at rest from a point \(O\). The acceleration of \(P\) at time \(t\) s after leaving \(O\) is denoted by \(a \text{ m s}^{-2}\), where $$a = 0.3t^{\frac{1}{2}} \quad \text{for } 0 \leqslant t \leqslant 4,$$ $$a = -kt^{-\frac{1}{2}} \quad \text{for } 4 < t \leqslant T,$$ where \(k\) and \(T\) are constants.

1. Find the velocity of \(P\) at \(t = 4\). [2]

1. It is given that there is no change in the velocity of \(P\) at \(t = 4\) and that the velocity of \(P\) at \(t = 16\) is \(0.3 \text{ m s}^{-1}\). Show that \(k = 2.6\) and find an expression, in terms of \(t\), for the velocity of \(P\) for \(4 \leqslant t \leqslant T\). [4]

1. Given that \(P\) comes to instantaneous rest at \(t = T\), find the exact value of \(T\). [2]

1. Find the total distance travelled between \(t = 0\) and \(t = T\). [4]

A particle \(X\) travels in a straight line. The velocity of \(X\) at time \(t\) s after leaving a fixed point \(O\) is denoted by \(v\) m s\(^{-1}\), where $$v = -0.1t^3 + 1.8t^2 - 6t + 5.6.$$ The acceleration of \(X\) is zero at \(t = p\) and \(t = q\), where \(p < q\).

1. Find the value of \(p\) and the value of \(q\). [4]

It is given that the velocity of \(X\) is zero at \(t = 14\).

1. Find the velocities of \(X\) at \(t = p\) and at \(t = q\), and hence sketch the velocity-time graph for the motion of \(X\) for \(0 \leq t \leq 15\). [3]
2. Find the total distance travelled by \(X\) between \(t = 0\) and \(t = 15\). [5]

A particle \(P\) moves in a straight line, passing through a point \(O\) with velocity \(4.2 \text{ ms}^{-1}\). At time \(t\) s after \(P\) passes \(O\), the acceleration, \(a \text{ ms}^{-2}\), of \(P\) is given by \(a = 0.6t - 2.7\). Find the distance \(P\) travels between the times at which it is at instantaneous rest. [7]

A particle moves in a straight line. It starts from rest, at time \(t = 0\), and accelerates at 0.6 t ms\(^{-2}\) for 4 s, reaching a speed of \(V\) ms\(^{-1}\). The particle then travels at \(V\) ms\(^{-1}\) for 11 s, and finally slows down, with constant deceleration, stopping after a further 5 s.

1. Show that \(V = 4.8\). [1]
2. Sketch a velocity-time graph for the motion. [3]
3. Find an expression, in terms of \(t\), for the velocity of the particle for \(15 \leqslant t \leqslant 20\). [2]
4. Find the total distance travelled by the particle. [4]

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

1. Find an expression, in terms of \(t\), for the displacement of \(P\) from \(O\). [4]
2. Find the velocity of \(P\) when its displacement from \(O\) is \(11.25 \text{ m}\). [3]

A particle \(P\) travels in a straight line from \(A\) to \(D\), passing through the points \(B\) and \(C\). For the section \(AB\) the velocity of the particle is \((0.5t - 0.01t^2)\) m s\(^{-1}\), where \(t\) s is the time after leaving \(A\).

1. Given that the acceleration of \(P\) at \(B\) is 0.1 m s\(^{-2}\), find the time taken for \(P\) to travel from \(A\) to \(B\). [3]

The acceleration of \(P\) from \(B\) to \(C\) is constant and equal to 0.1 m s\(^{-2}\).

1. Given that \(P\) reaches \(C\) with speed 14 m s\(^{-1}\), find the time taken for \(P\) to travel from \(B\) to \(C\). [3]

\(P\) travels with constant deceleration 0.3 m s\(^{-2}\) from \(C\) to \(D\). Given that the distance \(CD\) is 300 m, find

1. the speed with which \(P\) reaches \(D\), [2]
2. the distance \(AD\). [6]

A vehicle is moving in a straight line. The velocity \(v\) m s\(^{-1}\) at time \(t\) s after the vehicle starts is given by $$v = A(t - 0.05t^2) \quad \text{for } 0 \leq t \leq 15,$$ $$v = \frac{B}{t^2} \quad \text{for } t \geq 15,$$ where \(A\) and \(B\) are constants. The distance travelled by the vehicle between \(t = 0\) and \(t = 15\) is 225 m.

1. Find the value of \(A\) and show that \(B = 3375\). [5]
2. Find an expression in terms of \(t\) for the total distance travelled by the vehicle when \(t \geq 15\). [3]
3. Find the speed of the vehicle when it has travelled a total distance of 315 m. [3]

A particle moves in a straight line. It starts from rest at a fixed point \(O\) on the line. Its acceleration at time \(t\) s after leaving \(O\) is \(a\) m s\(^{-2}\), where \(a = 0.4t^3 - 4.8t^2\).

1. Show that, in the subsequent motion, the acceleration of the particle when it comes to instantaneous rest is \(16\) m s\(^{-2}\). [6]
2. Find the displacement of the particle from \(O\) at \(t = 5\). [3]

A particle starts from rest and moves in a straight line. The velocity of the particle at time \(t\) s after the start is \(v\) m s\(^{-1}\), where $$v = -0.01t^3 + 0.22t^2 - 0.4t.$$

1. Find the two positive values of \(t\) for which the particle is instantaneously at rest. [2]
2. Find the time at which the acceleration of the particle is greatest. [3]
3. Find the distance travelled by the particle while its velocity is positive. [4]

A particle moves in a straight line starting from a point \(O\) with initial velocity \(1\) m s\(^{-1}\). The acceleration of the particle at time \(t\) s after leaving \(O\) is \(a\) m s\(^{-2}\), where $$a = 1.2t^{\frac{1}{2}} - 0.6t.$$

1. At time \(T\) s after leaving \(O\) the particle reaches its maximum velocity. Find the value of \(T\). [2]
2. Find the velocity of the particle when its acceleration is maximum (you do not need to verify that the acceleration is a maximum rather than a minimum). [6]

A particle moves in a straight line, starting from rest at a point \(O\), and comes to instantaneous rest at a point \(P\). The velocity of the particle at time \(t\) s after leaving \(O\) is \(v\) m s\(^{-1}\), where $$v = 0.6t^2 - 0.12t^3.$$

1. Show that the distance \(OP\) is 6.25 m. [5]

On another occasion, the particle also moves in the same straight line. On this occasion, the displacement of the particle at time \(t\) s after leaving \(O\) is \(s\) m, where $$s = kt^3 + ct^5.$$ It is given that the particle passes point \(P\) with velocity 1.25 m s\(^{-1}\) at time \(t = 5\).

1. Find the values of the constants \(k\) and \(c\). [5]

1. Find the acceleration of the particle at time \(t = 5\). [2]

A particle moves in a straight line. At time \(t\) seconds its velocity is \(v\) ms\(^{-1}\) and its acceleration is \(a\) ms\(^{-2}\).

1. Given that \(a = —\), express \(v\) in terms of \(t\).
2. Given that \(v = tv\) when \(t = 0\), find \(v\) in terms of \(t\).
3. Find the displacement from the starting point when \(t = v\).

[6]

The position vector of a particle at time \(t\) is given by \(\mathbf{r} = t^2\mathbf{i} + (3t - 1)\mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. Find the velocity and acceleration of the particle when \(t = 2\).

1. Hence find the angle between the velocity and acceleration vectors when \(t = 2\). [3]
2. Find the value of \(t\) for which the velocity and acceleration vectors are perpendicular. [4]

A particle of mass \(2\) kg moves under the action of a variable force. At time \(t\) seconds the force is \((6t - 3)\mathbf{i} + 4\mathbf{j}\) newtons, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. When \(t = 0\), the particle is at rest at the origin.

1. Find the velocity of the particle when \(t = 4\). [4]
2. Find the kinetic energy of the particle when \(t = 4\). [2]
3. Find the distance of the particle from the origin when \(t = 2\). [6]

The equation of the trajectory of a small ball \(B\) projected from a fixed point \(O\) is $$y = -0.05x^2,$$ where \(x\) and \(y\) are, respectively, the displacements in metres of \(B\) from \(O\) in the horizontal and vertically upwards directions.

1. Show that \(B\) is projected horizontally, and find its speed of projection. [3]
2. Find the value of \(y\) when the direction of motion of \(B\) is \(60°\) below the horizontal, and find the corresponding speed of \(B\). [6]

\(O\), \(A\) and \(B\) are three points in a straight line on a smooth horizontal surface. A particle \(P\) of mass \(0.6\) kg moves along the line. At time \(t\) s the particle has displacement \(x\) m from \(O\) and speed \(v\) m s\(^{-1}\). The only horizontal force acting on \(P\) has magnitude \(0.4v^{\frac{1}{2}}\) N and acts in the direction \(OA\). Initially the particle is at \(A\), where \(x = 1\) and \(v = 1\).

1. Show that \(3v^{\frac{1}{2}}\frac{dv}{dx} = 2\). [2]
2. Express \(v\) in terms of \(x\). [4]
3. Given that \(AB = 7\) m, find the value of \(t\) when \(P\) passes through \(B\). [3]

A particle \(P\) moves in a straight line and passes through a point \(O\) of the line with velocity \(2\text{ m s}^{-1}\). At time \(t\) s after passing through \(O\), the velocity of \(P\) is \(v\text{ m s}^{-1}\) and the acceleration of \(P\) is given by \(\text{e}^{-0.5t}\text{ m s}^{-2}\). Calculate the velocity of \(P\) when \(t = 1.2\). [4]

A particle \(P\) of mass \(2\) kg moving on a horizontal straight line has displacement \(x\) m from a fixed point \(O\) on the line and velocity \(v\) m s\(^{-1}\) at time \(t\) s. The only horizontal force acting on \(P\) has magnitude \(\frac{1}{10}(2v - 1)^2 e^{-t}\) N and acts towards \(O\). When \(t = 0\), \(x = 1\) and \(v = 3\).

1. Find an expression for \(v\) in terms of \(t\). [5]
2. Find an expression for \(x\) in terms of \(t\). [4]

A particle \(P\) moving in a straight line has displacement \(x\) m from a fixed point \(O\) on the line at time \(t\) s. The acceleration of \(P\), in m s\(^{-2}\), is given by \(\frac{200}{x^2} - \frac{100}{x^3}\) for \(x > 0\). When \(t = 0\), \(x = 1\) and \(P\) has velocity \(10\) m s\(^{-1}\) directed towards \(O\).

1. Show that the velocity \(v\) m s\(^{-1}\) of \(P\) is given by \(v = \frac{10(1-2x)}{x}\). [5]
2. Show that \(x\) and \(t\) are related by the equation \(e^{-40t} = (2x-1)e^{2x-2}\) and deduce what happens to \(x\) as \(t\) becomes large. [5]

A particle \(P\) of mass \(m\) kg moves along a horizontal straight line with acceleration \(a\) ms\(^{-2}\) given by $$a = \frac{v(1-2t^2)}{t},$$ where \(v\) ms\(^{-1}\) is the velocity of \(P\) at time \(t\) s.

1. Find an expression for \(v\) in terms of \(t\) and an arbitrary constant. [3]
2. Given that \(a = 5\) when \(t = 1\), find an expression, in terms of \(m\) and \(t\), for the horizontal force acting on \(P\) at time \(t\). [3]

A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v\) m s\(^{-1}\) in the direction of \(x\) increasing, where $$v = (t - 2)(3t - 10), \quad t \geq 0$$ When \(t = 0\), \(P\) is at the origin \(O\).

1. Find the acceleration of \(P\) when \(t = 3\) [3]
2. Find the total distance travelled by \(P\) in the first 3 seconds of its motion. [6]
3. Show that \(P\) never returns to \(O\). [2]