Description
1. A rocket is fired horizontally off of a rooftop. As it leaves the rooftop it has an
initial horizontal velocity v0 and a constant horizontal acceleration a0 in addition to
the acceleration, g, downward due to gravity. What is the shape of its trajectory?
(Hyperbola? Parabola? Straight line? Or something else?)
Hint: This question can
be answered without any need for calculation, if you think geometrically).
2. Byron and Fuller, Chapter 1, problem 1
3. Show that ijkijk = 6.
4. Demonstrate algebraically whether or not the cross product is associative. That is,
verify or falsify the following:
a × (b × c) = (a × b) × c
If they are not in general equal, are they at least of equal magnitude?
5. Consider a two dimensional system where the vector ~x is given by
~x = x1eˆ1 + x2eˆ2
and the x-coordinate is transformed into a different, non-orthogonal coordinates as:
x
0
1 =
1
√
2
(x1 + x2)
x
0
2 = x2
The gradient of the scalar function is:
∇~ φ(~x) = ∂1φ eˆ1 + ∂2φ eˆ2
Find the components of the gradient in the primed co-ordinates and show that it
transforms as a covariant vector.
6. Show that for an orthogonal transformation, there is no distinction between a contravariant and a covariant vector.
Figure 1: Cartoon by R. Bolling, ‘Super-Fun-Pak Comix”.
