Bullet Ballistics - Calculating Minimum Muzzle Velocity

You're one crime scene away from the weekend.


In this scenario, you find a 8.2 gram lead bullet that was stopped in a doorframe. It's melted completely on impact. Assuming the bullet was shot at room temperature (20° C), what do you calculate as the minimum speed the bullet leaves the gun (aka its minimum muzzle velocity)?

Note: Let's say lead melts at 327°C, it's specific heat capacity is 130 J/kg * °C, and its specific latent heat of fusion (melting) is 2.5 * 10^{4} J/kg.

(Stuck? Check the hint)

      223 m/s
      283 m/s
      360 m/s

(Calculus) Angle Between Two Vectors

The angle in  degrees between two vectors 


most nearly is 

      A) 8.124
      B) 11.47
      C) 78.52
      D) 81.88

(Original question contributed by Professor Autar Kaw. Modified with permission.)

(Math) Ever Try Bungee Jumping?

Bungee jumping may seem like a reckless activity to some, but in truth, many commercial bungee operators rigorously ensure their equipment conforms to industry standards and guidelines. Operators commonly double and triple check the calculations and fittings before every jump.

As an operator, you need to determine the appropriate rope length, which may differ from person to person, day to day (weather conditions), and rope to rope ("stretchiness").

To prevent whiplash, injuries (and a subsequent lawsuit), bungee operators typically aim for the rope to elongate to an additional 300% (i.e. x4 normal length) for a softer fall.

If your 5'5" 145 lb girlfriend/boyfriend/malamut wants to jump and just barely glance a 47m lake below, what length should you spec the rope to be?

Stuck? Click here for a hint!
      6.5 m
      8.25 m
      11.35 m
(The founder Jen Tran actually jumped from the lake pictured above in the video.)

(Math) Discovering Pi

To find the value of π, a scientist inscribes (bakes) a n-sided polygon (pie) in a circle (pan) of diameter 1. The perimeter of the regular polygon is the value of π for n --> ∞ 

The approximate value of π by using a 6-sided regular polygon is...

Stuck? Click here for a hint!

(Original question contributed by Professor Autar Kaw; modified under permission of CCL)

(Calculus) Revolve Them Ounces Off the Shuttle

You’re designing the structural component below using CAD (Computer-Aided-Design). It's sketched out in 2D, but you're going to revolve it around the horizontal axis to make it a 3D part.

The current design (left side below) is a solid part made out of titanium (density = 4.506 g/cm^3). The part's going to go inside a space shuttle. To decrease material cost and overall lift-off weight of the shuttle, you decide to see how much mass you can shave off (right side below).

The part has a horizontal length of 10 m. After going through structural analysis of several hollowed versions, the profile above seems to meet design specifications of cost and weight reduction.

How much mass did you shave off from the original design?

Stuck? Click here for a hint!