6.03k Newton's experimental law: direct impact

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Pre-U Pre-U 9794/3 2016 June Q10
7 marks Standard +0.3
  1. A particle \(A\) of mass \(m\) travelling with speed \(u\) on a smooth horizontal surface collides directly with a particle \(B\) of mass \(3m\) travelling with speed \(\frac{2u}{5}\) in the opposite direction. After the collision, \(A\) travels at speed \(\frac{2u}{5}\) and \(B\) travels at speed \(\frac{4u}{15}\), both in the same direction as \(B\) before the collision. Find \(A\) and the coefficient of restitution between the two particles. [4]
  2. A particle of mass 3 kg moving with velocity \((2\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}) \text{ m s}^{-1}\) receives an impulse of \((6\mathbf{i} - 6\mathbf{j} - 9\mathbf{k})\) N s. Find the velocity of the particle after the impulse. [3]
Pre-U Pre-U 9794/3 2019 Specimen Q9
10 marks Challenging +1.2
\includegraphics{figure_9} Three particles \(A\), \(B\) and \(C\), having masses of 1 kg, 2 kg and 5 kg respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed 14 m s\(^{-1}\). The coefficient of restitution between each pair of particles is 0.5.
  1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest. [4]
  2. Show that \(B\) reversed direction after the impact with \(C\). [3]
  3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time. [3]
Pre-U Pre-U 9794/3 2020 Specimen Q9
10 marks Challenging +1.2
\includegraphics{figure_9} Three particles \(A\), \(B\) and \(C\), having masses of 1 kg, 2 kg and 5 kg respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed 14 ms\(^{-1}\). The coefficient of restitution between each pair of particles is 0.5.
  1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest. [4]
  2. Show that \(B\) reversed direction after the impact with \(C\). [3]
  3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time. [3]
Pre-U Pre-U 9794/2 Specimen Q10
7 marks Moderate -0.3
  1. Determine the impulse of a force of magnitude \(6\) N that acts on a particle of mass \(3\) kg for \(1.5\) seconds. [1]
Particles \(A\) and \(B\), of masses \(0.1\) kg and \(0.2\) kg respectively, can move on a smooth horizontal table. Initially \(A\) is moving with speed \(3\) m s\(^{-1}\) towards \(B\), which is moving with speed \(1\) m s\(^{-1}\) in the same direction as the motion of \(A\). During a collision \(B\) experiences an impulse from \(A\) of magnitude \(0.2\) kg m s\(^{-1}\).
  1. Find the speeds of the particles immediately after the collision. [4]
  2. Determine the coefficient of restitution between the particles. [2]
CAIE FP2 2013 November Q11
Challenging +1.8
11 Answer only one of the following two alternatives.
EITHER
A smooth sphere, with centre \(O\) and radius \(a\), is fixed on a smooth horizontal plane \(\Pi\). A particle \(P\) of mass \(m\) is projected horizontally from the highest point of the sphere with speed \(\sqrt { } \left( \frac { 2 } { 5 } g a \right)\). While \(P\) remains in contact with the sphere, the angle between \(O P\) and the upward vertical is denoted by \(\theta\). Show that \(P\) loses contact with the sphere when \(\cos \theta = \frac { 4 } { 5 }\). Subsequently the particle collides with the plane \(\Pi\). The coefficient of restitution between \(P\) and \(\Pi\) is \(\frac { 5 } { 9 }\). Find the vertical height of \(P\) above \(\Pi\) when the vertical component of the velocity of \(P\) first becomes zero.
OR
A factory produces bottles of spring water. The manager decides to assess the performance of the two machines that are used to fill the bottles with water. He selects a random sample of 60 bottles filled by the first machine \(X\) and a random sample of 80 bottles filled by the second machine \(Y\). The volumes of water, \(x\) and \(y\), measured in appropriate units, are summarised as follows. $$\Sigma x = 58.2 \quad \Sigma x ^ { 2 } = 85.8 \quad \Sigma y = 97.6 \quad \Sigma y ^ { 2 } = 188.6$$ A test at the \(\alpha \%\) significance level shows that the mean volume of water in bottles filled by machine \(X\) is less than the mean volume of water in bottles filled by machine \(Y\). Find the set of possible values of \(\alpha\).