Course:PHYS350/2007W Midterm 1

1. A ball is dropped from height $h$ and bounces. The coefficient of restitution at each bounce is $e$ . Find the velocity immediately after the first bounce, and immediately after the nth bounce. Find how long it takes for the ball to finally come to rest.

Let's calculate its velocity when it first hits. We have

$m g h = \frac{1}{2} m v^{2}$

so

$v = \sqrt{2 g h} .$

After it bounces it will have a velocity of $e v_{0}$ so $v_{n} = e^{n} \sqrt{2 g h}$ . How long does each bounce last? The initial velocity is $v_{n} = e^{n} \sqrt{2 g h}$ up and the velocity just before the next bounce is $v_{n}$ down, so each bounce lasts

$t_{n} = \frac{2 v_{n}}{g} = 2 e^{n} \sqrt{\frac{2 h}{g}} .$

Let's sum over the bounces and the initial drop

$t = \sqrt{\frac{2 h}{g}} + 2 \sqrt{\frac{2 h}{g}} \sum_{n = 1}^{\infty} e^{n} = (1 + \frac{2 e}{1 - e}) \sqrt{\frac{2 h}{g}} = \frac{1 + e}{1 - e} \sqrt{\frac{2 h}{g}}$

2. A particle of mass $m$ slides without friction on the inside of a spherical shell of radius $R$ . What is the Lagrangian for this system including a uniform gravitational field? What are the equations of motion for the mass? What quantities are conserved?

It is natural to use spherical polar coordinates for this, so we have $z = - R \cos θ$ so

$V = - m R \cos θ .$

The kinetic energy is

$T = \frac{1}{2} m R^{2} (\sin^{2} θ {\dot{ϕ}}^{2} + {\dot{θ}}^{2})$

so

$L = \frac{1}{2} m R^{2} (\sin^{2} θ {\dot{ϕ}}^{2} + {\dot{θ}}^{2}) + m R \cos θ .$

And for the equations of motion,

$m R^{2} \ddot{θ} = - m R \sin θ + m R^{2} \sin θ \cos θ$

and

$\frac{d}{d t} (m R^{2} \sin^{2} θ \dot{ϕ}) = 0$

where the quantity in parenthesis is the conserved angular momentum. The energy $T + V$ is also conserved.