Displacement from velocity by integration

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 stone is thrown vertically upwards with speed \(16 \text{ m s}^{-1}\) from a point \(h\) metres above the ground. The stone hits the ground \(4\) s later. Find

1. the value of \(h\), [3]
2. the speed of the stone as it hits the ground. [3]

A ball is projected vertically upwards with speed 21 m s\(^{-1}\) from a point \(A\), which is 1.5 m above the ground. After projection, the ball moves freely under gravity until it reaches the ground. Modelling the ball as a particle, find

1. the greatest height above \(A\) reached by the ball, [3]
2. the speed of the ball as it reaches the ground, [3]
3. the time between the instant when the ball is projected from \(A\) and the instant when the ball reaches the ground. [4]

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 \(v = 6t - 2t^2\). When \(t = 0\), \(P\) is at the origin \(O\). Find the distance of \(P\) from \(O\) when \(P\) comes to instantaneous rest after leaving \(O\). [5]

\includegraphics{figure_5} A toy car is moving along the straight line \(Ox\), where O is the origin. The time \(t\) is in seconds. At time \(t = 0\) the car is at A, 3 m from O as shown in Fig. 5. The velocity of the car, \(v\) m s\(^{-1}\), is given by $$v = 2 + 12t - 3t^2.$$ Calculate the distance of the car from O when its acceleration is zero. [8]

A remote-controlled toy car is moving over a horizontal surface. It moves in a straight line through a point \(A\). The toy is initially at the point with displacement \(3\) metres from \(A\). Its velocity, \(v\,\mathrm{m}\,\mathrm{s}^{-1}\), at time \(t\) seconds is defined by $$v = 0.06(2 + t - t^2)$$

1. Find an expression for the displacement, \(r\) metres, of the toy from \(A\) at time \(t\) seconds. [4 marks]
2. In this question use \(g = 9.8\,\mathrm{m}\,\mathrm{s}^{-2}\) At time \(t = 2\) seconds, the toy launches a ball which travels directly upwards with initial speed \(3.43\,\mathrm{m}\,\mathrm{s}^{-1}\) Find the time taken for the ball to reach its highest point. [3 marks]

A particle, of mass 400 grams, is initially at rest at the point \(O\). The particle starts to move in a straight line so that its velocity, \(v\) m s⁻¹, at time \(t\) seconds is given by \(v = 6t^2 - 12t^3\) for \(t > 0\)

1. Find an expression, in terms of \(t\), for the force acting on the particle. [3 marks]
2. Find the time when the particle next passes through \(O\). [5 marks]

An elite athlete runs in a straight line to complete a 100-metre race. During the race, the athlete's velocity, \(v \text{ m s}^{-1}\), may be modelled by $$v = 11.71 - 11.68e^{-0.9t} - 0.03e^{0.3t}$$ where \(t\) is the time, in seconds, after the starting pistol is fired.

1. Find the maximum value of \(v\), giving your answer to one decimal place. Fully justify your answer. [8 marks]
2. Find an expression for the distance run in terms of \(t\). [6 marks]
3. The athlete's actual time for this race is 9.8 seconds. Comment on the accuracy of the model. [2 marks]

A particle moves along the horizontal \(x\)-axis so that its velocity \(v\) ms\(^{-1}\) at time \(t\) seconds is given by $$v = 6t^2 - 8t - 5.$$ At time \(t = 1\), the particle's displacement from the origin is \(-4\) m. Find an expression for the displacement of the particle at time \(t\) seconds. [3]

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]