Work done by non-constant force integration

2. A particle of mass 4 kg is moving along the horizontal \(x\)-axis under the action of a single force which acts in the positive \(x\)-direction. At time \(t\) seconds the force has magnitude \(\left( 1 + 3 t ^ { \frac { 1 } { 2 } } \right) \mathrm { N }\).
When \(t = 0\) the particle has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Find the work done by the force in the interval \(0 \leqslant t \leqslant 4\)

2 An object moves under the action of a single force \(F\) newtons.
It is given that \(F = 6 x ^ { 2 }\), where \(x\) represents the displacement in metres from the initial position of the object. Find the work done by \(F\) in moving the object from \(x = 1\) to \(x = 2\) Circle your answer.
[0pt] [1 mark]
12 J
14 J
18J
42 J

1 A particle moves along the \(x\)-axis under the action of a force, \(F\) newtons, where $$F = 3 x ^ { 2 } + 5$$ Find the work done by the force as the particle moves from \(x = 0\) metres to \(x = 2\) metres. Circle your answer.
12 J
17 J
18 J
34 J

2 The tension, \(T\) newtons, in a spring is given by \(T = 20 e\), where \(e\) metres is the extension of the spring. Calculate the work done when the extension is increased from 0.2 metres to 0.4 metres. Circle your answer.
[0pt] [1 mark]
0.4 J 0.9 J 1.2 J 1.6 J

5 A train of mass 10000 kg is travelling at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides with a buffer. The buffer brings the train to rest. As the buffer brings the train to rest it compresses by 0.2 metres.
When the buffer is compressed by a distance of \(x\) metres it exerts a force of magnitude \(F\) newtons, where $$F = A x + 9000 x ^ { 2 }$$ where \(A\) is a constant. 5

1. Find, in terms of \(A\), the work done in compressing the buffer by 0.2 metres.
   5
2. Find the value of \(A\)