Elastic collisions in two dimensions: แบบฝึกหัด

A smooth sphere of mass \(m\) moves with speed \(u\) on a smooth horizontal floor. It strikes a smooth vertical wall at an angle \(\alpha\) to the normal. The coefficient of restitution between the sphere and the wall is \(e\). If the kinetic energy of the sphere after the impact is half of its kinetic energy before the impact, which of the following equations must hold?

A smooth sphere of mass \(m\) moving with velocity \((3\mathbf{i} + 4\mathbf{j})\) ms\(^{-1}\) collides with a smooth fixed vertical wall. The wall lies in the plane defined by the vector \(\mathbf{j}\). The coefficient of restitution is \(e = 0.6\). Calculate the vector impulse exerted by the wall on the sphere.

A smooth sphere \(A\) of mass \(m\) moving with velocity \(u\) hits an identical smooth sphere \(B\) of mass \(m\) which is at rest. The velocity of \(A\) makes an angle of \(30^\circ\) with the line of centers at the moment of impact. The coefficient of restitution between the spheres is \(e = \frac{1}{2}\). Find the speed of sphere \(B\) immediately after the impact.

A smooth sphere \(P\) strikes an identical smooth sphere \(Q\) which is at rest. Before the collision, \(P\) is moving at a speed \(V\) in a direction making an angle \(\theta\) with the line of centers. The coefficient of restitution is \(e\). Show that the angle \(\phi\) through which the direction of motion of \(P\) is deflected is given by \(\tan\phi = \frac{(1+e)\tan\theta}{2 - (1-e)\tan^2\theta}\)? No, calculate the component of velocity of \(P\) perpendicular to the line of centers after impact.

Two identical smooth spheres \(A\) and \(B\) are moving on a smooth horizontal surface. \(A\) has velocity \(u\mathbf{i}\) and \(B\) has velocity \(-u\mathbf{j}\). At the moment of impact, the line of centers is parallel to \(\mathbf{i}\). If the collision is perfectly elastic (\(e=1\)), what are the velocities of \(A\) and \(B\) after impact?

A smooth sphere of mass \( m \) moves with speed \( u \) on a horizontal plane and strikes a fixed smooth vertical wall. The direction of motion of the sphere before impact makes an angle \( \alpha \) with the wall. Given the coefficient of restitution between the sphere and the wall is \( e \), find the magnitude of the velocity component of the sphere parallel to the wall after impact.

Two identical smooth spheres \( A \) and \( B \) are moving on a smooth horizontal surface. \( A \) has velocity \( (3ι + 2ϊ) \text{ m s}^{-1} \) and \( B \) has velocity \( (2ι - ϊ) \text{ m s}^{-1} \). At the instant of collision, the line of centres is parallel to \( ι \). If \( e = 0.5 \), calculate the velocity vector of sphere \( A \) after the collision.

A smooth sphere collides obliquely with a fixed smooth plane. Its velocity before impact is \( \mathbf{u} = 4\mathbf{i} + 3\mathbf{j} \) and its velocity after impact is \( \mathbf{v} = 4\mathbf{i} - 1.5\mathbf{j} \), where \( \mathbf{i} \) is parallel to the plane and \( \mathbf{j} \) is perpendicular to the plane. Determine the coefficient of restitution \( e \) between the sphere and the plane.

A smooth sphere \( S \) of mass \( m \) is moving on a smooth horizontal floor with velocity \( (3\mathbf{i} + 4\mathbf{j}) \text{ m s}^{-1} \). It strikes a smooth vertical wall which lies in the plane of the vector \( \mathbf{i} \). The coefficient of restitution between the sphere and the wall is \( e = 0.5 \).

(a) Find the velocity of the sphere immediately after the impact.

(b) Calculate the loss in kinetic energy of the sphere due to the impact.

A particle \( P \) strikes a smooth fixed plane at an angle \( \theta \) to the normal. The coefficient of restitution between the particle and the plane is \( e \). After the impact, the particle moves at an angle \( \phi \) to the normal.

(a) Prove that \( \tan \phi = \frac{1}{e} \tan \theta \).

(b) If the kinetic energy of the particle is halved by the impact, show that \( e^2 = \frac{1 - 2\sin^2\theta}{2\cos^2\theta} \), and deduce the range of values of \( \theta \) for which this is possible.