3.02f Non-uniform acceleration: using differentiation and integration

3 A particle moves in a straight line and at time \(t\) has velocity \(v\), where $$v = 2 t - 12 \mathrm { e } ^ { - t } , \quad t \geqslant 0$$

1. 1. Find an expression for the acceleration of the particle at time \(t\).
   2. State the range of values of the acceleration of the particle.
2. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).
   (4 marks)

5 Tom is on a fairground ride.
Tom's position vector, \(\mathbf { r }\) metres, at time \(t\) seconds is given by $$\mathbf { r } = 2 \cos t \mathbf { i } + 2 \sin t \mathbf { j } + ( 10 - 0.4 t ) \mathbf { k }$$ The perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the horizontal plane and the unit vector \(\mathbf { k }\) is directed vertically upwards.

1. 1. Find Tom's position vector when \(t = 0\).
   2. Find Tom's position vector when \(t = 2 \pi\).
   3. Write down the first two values of \(t\) for which Tom is directly below his starting point.
2. Find an expression for Tom's velocity at time \(t\).
3. Tom has mass 25 kg . Show that the resultant force acting on Tom during the motion has constant magnitude. State the magnitude of the resultant force.
   (5 marks)

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 \(t\).

4 A particle moves so that at time \(t\) seconds its velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) is given by $$\mathbf { v } = \left( 4 t ^ { 3 } - 12 t + 3 \right) \mathbf { i } + 5 \mathbf { j } + 8 t \mathbf { k }$$

1. When \(t = 0\), the position vector of the particle is \(( - 5 \mathbf { i } + 6 \mathbf { k } )\) metres. Find the position vector of the particle at time \(t\).
2. Find the acceleration of the particle at time \(t\).
3. Find the magnitude of the acceleration of the particle at time \(t\). Do not simplify your answer.
4. Hence find the time at which the magnitude of the acceleration is a minimum.
5. The particle is moving under the action of a single variable force \(\mathbf { F }\) newtons. The mass of the particle is 7 kg . Find the minimum magnitude of \(\mathbf { F }\).

1 A particle moves in a straight line and at time \(t\) seconds has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 6 t ^ { 2 } + 4 t - 7 , \quad t \geqslant 0$$

1. Find an expression for the acceleration of the particle at time \(t\).
2. The mass of the particle is 3 kg . Find the resultant force on the particle when \(t = 4\).
3. When \(t = 0\), the displacement of the particle from the origin is 5 metres. Find an expression for the displacement of the particle from the origin at time \(t\).

13 A ball falls freely towards the Earth.
The ball passes through two different fixed points \(M\) and \(N\) before reaching the Earth's surface. At \(M\) the ball has velocity \(u \mathrm {~ms} ^ { - 1 }\) At \(N\) the ball has velocity \(3 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) It can be assumed that:

• the motion is due to gravitational force only
• the acceleration due to gravity remains constant throughout.

13

1. Show that the time taken for the ball to travel from \(M\) to \(N\) is \(\frac { 2 u } { g }\) seconds.
   [0pt] [2 marks] 13
2. Point \(M\) is \(h\) metres above the Earth. Show that \(h > \frac { 4 u ^ { 2 } } { g }\) Fully justify your answer.
   The car is moving in a straight line.
   The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the car at time \(t\) seconds is given by $$a = 3 k t ^ { 2 } - 2 k t + 1$$ where \(k\) is a constant.
   When \(t = 3\) the car has a velocity of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Show that \(k = \frac { 1 } { 3 }\)

A particle, \(P\), moves along the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the displacement, \(x\) metres, of \(P\) from the origin \(O\), is given by \(x = \frac { 1 } { 2 } t ^ { 2 } \left( t ^ { 2 } - 2 t + 1 \right)\)

1. Find the times when \(P\) is instantaneously at rest.
2. Find the total distance travelled by \(P\) in the time interval \(0 \leqslant t \leqslant 2\)
3. Show that \(P\) will never move along the negative \(x\)-axis.

1. A bird leaves its nest at time \(t = 0\) for a short flight along a straight line.

The bird then returns to its nest.
The bird is modelled as a particle moving in a straight horizontal line.
The distance, \(s\) metres, of the bird from its nest at time \(t\) seconds is given by $$s = \frac { 1 } { 10 } \left( t ^ { 4 } - 20 t ^ { 3 } + 100 t ^ { 2 } \right) , \quad \text { where } 0 \leqslant t \leqslant 10$$

1. Explain the restriction, \(0 \leqslant t \leqslant 10\)
2. Find the distance of the bird from the nest when the bird first comes to instantaneous rest.

6. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) moves in the \(x - y\) plane in such a way that its velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) is given by $$\mathbf { v } = t ^ { - \frac { 1 } { 2 } } \mathbf { i } - 4 \mathbf { j }$$ When \(t = 1 , P\) is at the point \(A\) and when \(t = 4 , P\) is at the point \(B\).
Find the exact distance \(A B\).

6. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) moves so that its acceleration \(\mathbf { a } \mathrm { m } \mathrm { s } ^ { - 2 }\) is given by $$\mathbf { a } = 5 t \mathbf { i } - 15 t ^ { \frac { 1 } { 2 } } \mathbf { j }$$ When \(t = 0\), the velocity of \(P\) is \(20 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the speed of \(P\) when \(t = 4\)

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) is moving with velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where $$\mathbf { v } = 3 ( t + 2 ) ^ { 2 } \mathbf { i } + 5 t ( t + 2 ) \mathbf { j }$$ Position vectors are given relative to the fixed point \(O\) At time \(t = 0 , P\) is at the point with position vector \(( - 30 \mathbf { i } - 45 \mathbf { j } ) \mathrm { m }\).

1. Find the position vector of \(P\) when \(t = 3\)
2. Find the magnitude of the acceleration of \(P\) when \(t = 3\) At time \(T\) seconds, \(P\) is moving in the direction of the vector \(2 \mathbf { i } + \mathbf { j }\)
3. Find the value of \(T\)

6 A particle travels along a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after \(t\) seconds is given by $$v = t ^ { 3 } - 6 t ^ { 2 } + 8 t \text { for } 0 \leqslant t \leqslant 4$$ When \(t = 0\) the particle is at rest at the point \(P\).

1. Find the times (other than \(t = 0\) ) when the particle is at rest. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 4\).
2. Find the acceleration of the particle when \(t = 2\).
3. Find an expression for the displacement of the particle from \(P\) after \(t\) seconds. Hence state its displacement from \(P\) when \(t = 2\) and find its average speed between \(t = 0\) and \(t = 2\).

7 A particle travels along a straight line. Its velocity \(v \mathrm {~ms} ^ { - 1 }\) after \(t\) seconds is given by $$v = t ^ { 3 } - 9 t ^ { 2 } + 20 t$$ 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. Find the displacement of the particle from \(P\) when \(t = 2\).

9 A particle moves along a straight line such that its displacement from \(O\), a fixed point on the line, is \(x\). The particle travels from rest from the point \(P\), where \(x = 2\), to the point \(Q\), where \(x = 5.6\). All distances are in metres. Two models for the motion of the particle are proposed.

1. In Model 1, the acceleration of the particle is assumed to be constant and the particle takes 18 seconds to travel from \(P\) to \(Q\). Find the velocity of the particle when it reaches \(Q\).
2. In Model 2, the velocity after \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \frac { 1 } { 270 } \left( 18 t - t ^ { 2 } \right)\).
   1. Write down the values of \(t\) when \(v = 0\).
   2. Show that \(x = 5.6\) when \(t = 18\).
   3. The particle represents a fragile instrument that is being moved from \(P\) to \(Q\) across a laboratory. Explain why Model 2 might be more appropriate than Model 1.

10 A particle \(P\) moves in a straight line starting from \(O\). At time \(t\) seconds after leaving \(O\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 5 + 1.5 t - 0.125 t ^ { 3 }\).

1. Find the displacement of \(P\) between the times \(t = 1\) and \(t = 4\).
2. Find the time at which the velocity of \(P\) is a maximum, justifying your answer.

A particle \(P\) moves along the \(x\)-axis so that its velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) seconds \(( t \geqslant 0 )\) is given by $$v = 3 t ^ { 2 } - 24 t + 36$$ a) Find the values of \(t\) when \(P\) is instantaneously at rest.
b) Calculate the total distance travelled by the particle \(P\) whilst its velocity is decreasing.

\includegraphics{figure_10} A tank containing water is in the form of a cone with vertex \(C\). The axis is vertical and the semi-vertical angle is \(60°\), as shown in the diagram. At time \(t = 0\), the tank is full and the depth of water is \(H\). At this instant, a tap at \(C\) is opened and water begins to flow out. The volume of water in the tank decreases at a rate proportional to \(\sqrt{h}\), where \(h\) is the depth of water at time \(t\). The tank becomes empty when \(t = 60\).

1. Show that \(h\) and \(t\) satisfy a differential equation of the form $$\frac{dh}{dt} = -Ah^{-\frac{1}{2}},$$ where \(A\) is a positive constant. [4]
2. Solve the differential equation given in part (i) and obtain an expression for \(t\) in terms of \(h\) and \(H\). [6]
3. Find the time at which the depth reaches \(\frac{1}{2}H\). [1]

[The volume \(V\) of a cone of vertical height \(h\) and base radius \(r\) is given by \(V = \frac{1}{3}\pi r^2 h\).]

A particle moves in a straight line \(AB\). The velocity \(v\text{ m s}^{-1}\) of the particle \(t\text{ s}\) after leaving \(A\) is given by \(v = t(5 - 2t)\) where \(k\) is a constant. The displacement of the particle from \(A\), in the direction towards \(B\), is \(2.5\text{ m}\) when \(t = 3\) and is \(2.4\text{ m}\) when \(t = 6\).

1. Find the value of \(k\). Hence find an expression, in terms of \(t\), for the displacement of the particle from \(A\). [7]
2. Find the displacement of the particle from \(A\) when its velocity is a minimum. [4]

A particle \(P\) moves in a straight line. The velocity \(v \text{ ms}^{-1}\) at time \(t\) s is given by $$v = 2t + 1 \quad \text{for } 0 \leqslant t \leqslant 5,$$ $$v = 36 - t^2 \quad \text{for } 5 \leqslant t \leqslant 7,$$ $$v = 2t - 27 \quad \text{for } 7 \leqslant t \leqslant 13.5.$$

1. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 13.5\). [3]
2. Find the acceleration at the instant when \(t = 6\). [2]
3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 13.5\). [5]

A particle moving in a straight line starts from rest at a point \(A\) and comes instantaneously to rest at a point \(B\). The acceleration of the particle at time \(t\) s after leaving \(A\) is \(a \text{ m s}^{-2}\), where $$a = 6t^{\frac{1}{2}} - 2t.$$

1. Find the value of \(t\) at point \(B\). [3]
2. Find the distance travelled from \(A\) to the point at which the acceleration of the particle is again zero. [5]

A particle starts from a point \(O\) and moves in a straight line. The velocity \(v\) m s\(^{-1}\) of the particle at time \(t\) s after leaving \(O\) is given by $$v = k(3t^2 - 2t^3),$$ where \(k\) is a constant.

1. Verify that the particle returns to \(O\) when \(t = 2\). [4]
2. It is given that the acceleration of the particle is \(-13.5\) m s\(^{-2}\) for the positive value of \(t\) at which \(v = 0\). Find \(k\) and hence find the total distance travelled in the first two seconds of motion. [6]

A particle \(P\) moves in a straight line. The velocity \(v\text{ms}^{-1}\) at time \(t\) seconds is given by $$v = 0.5t \quad \text{for } 0 \leqslant t \leqslant 10,$$ $$v = 0.25t^2 - 8t + 60 \quad \text{for } 10 < t \leqslant 20.$$

1. Show that there is an instantaneous change in the acceleration of the particle at \(t = 10\). [3]
2. Find the total distance covered by \(P\) in the interval \(0 \leqslant t \leqslant 20\). [6]

A particle moves in a straight line starting from rest. The displacement \(s\) m of the particle from a fixed point \(O\) on the line at time \(t\) s is given by $$s = t^2 - \frac{15}{4}t^2 + 6.$$ Find the value of \(s\) when the particle is again at rest. [4]

A particle \(P\) moves in a straight line. At time \(t\) s after leaving a point \(O\) on the line, \(P\) has velocity \(v\text{ ms}^{-1}\), where \(v = 44t - 6t^2 - 36\).

1. Find the set of values of \(t\) for which the acceleration of the particle is positive. [2]
2. Find the two values of \(t\) at which \(P\) returns to \(O\). [3]

A particle moves in a straight line starting from rest from a point \(O\). The acceleration of the particle at time \(t\) s after leaving \(O\) is \(a\,\text{m}\,\text{s}^{-2}\), where \(a = 4t^2\).

1. Find the speed of the particle when \(t = 9\). [2]
2. Find the time after leaving \(O\) at which the speed (in metres per second) and the distance travelled (in metres) are numerically equal. [3]