Hollywood often portrays bows as extremely lethal. In reality, however, they were being replaced by firearms on the battlefields of Europe already towards the end of the 100 years’ war (1337-1453). While all early projectile weapons were very inaccurate (sights are a recent invention), the most striking difference between the bow and the musket lies in the kinetic energy carried by the projectile. A well trained archer using the famous British Longbow could fire an arrow of about 100 grams (0.1 kg) with an initial energy of approximately 100 J.

QUESTION 1: Compare this to a black-powder musket and a modern rifle. Flintlock muskets typically fired a 30 gram projectile with an initial velocity of 200-300 m/s, whereas a modern US M4 carbine fires a 4 gram bullet at 910 m/s.

QUESTION 2: What would the recoil momentum be for all three weapons?

QUESTION 3: If we assume that the longbow stores energy like a spring, what would the spring constant k be? For simplicity, assume that the displacement is equal to the length of the arrow, which was about 30 inches (0.76 m).

QUESTION 4: In movies and works of fiction skilled archers are sometimes portrayed releasing multiple arrows at the same time. A real archer would, however, never do this. Why?

A ring (mass 2 M, radius 1 R) rotates in a CCW direction with an initial angular speed 2 ω. A disk (mass 4 M, radius 2 R) rotates in a CW direction with initial angular speed 4 ω. The ring and disk "collide" and eventually rotate together. Assume that positive angular momentum and angular velocity values correspond to rotation in the CCW direction.

What is the initial angular momentum Li of the ring+disk system? Write your answer in terms of MR2ω.

What is the final angular velocity ω_f of the ring+disk system? Write your answer in terms of ω.

List the factors that comprise the image formation model. Describe each briefly.

1. Dark Energy

(a) Why is dark energy "dark"? Why is it "energy"?

(b) Give at least one way that dark energy is different from dark matter. Also give at least one way that dark energy is similar to dark matter.

(c) What is the difference between the cosmological constant and dark energy?

Jake asked Sally to draw a vector perpendicular to the vector 4.0i - 6.0j +2.0k. Which vector would Sally draw?

A) 1.0i-2.0j+1.0k B) 2.0i -3.0j +1.0k C) 2.0i+1.0k D) 1.0i +1.0j+1.0k E) 3.0i -3.0j+3.0k

A train car with mass m1 = 576 kg is moving to the right with a speed of v1 = 7 m/s and collides with a second train car. The two cars latch together during the collision and then move off to the right at vf = 4.4 m/s.

A)What is the initial momentum of the first train car?
B)What is the mass of the second train car?
C)What is the change in kinetic energy of the two train system during the collision?
D)Now the same two cars are involved in a second collision. The first car is again moving to the right with a speed of v₁ = 7 m/s and collides with the second train car that is now moving to the left with a velocity v₂ = -5.1 m/s before the collision. The two cars latch together at impact.

What is the final velocity of the two-car system? (A positive velocity means the two train cars move to the right – a negative velocity means the two train cars move to the left.)

An object 0.800 cm tall is placed 16.5 cm to the left of the vertex of a concave spherical mirror having a radius of curvature of 20.5 cm.

A:

Calculate the position of the image.

Express your answer in centimeters to three significant figures.

s'=?

B.

Calculate the size of the image.

Express your answer in centimeters to three significant figures.

'y'= ?

C.

Find the orientation (upright or inverted) and the nature (real or virtual) of the image.

Options:

upright and real

upright and virtual

inverted and real

inverted and virtual

Water is flowing in the pipe shown in the figure below, with the 7.70-cm diameter at point 1 tapering to 3.45 cm at point 2, located y = 11.5 cm below point 1. The water pressure at point 1 is 3.20 ✕ 104 Pa and decreases by 50% at point 2. Assume steady, ideal flow. What is the speed of the water at the following points? Point 1? Point 2?

A disk of radius 14 cm and has a mass of 10 kg is attached to a hub which is a
disk of mass 8 kg and radius 5cm. The disks are mounted on a frictionless axel
that runs through the center. A rope passes over the hub and supports a 10 kg
mass on one side and a 5 kg mass on the other. If the rope does not slip on the
hub,
a.) What is the tension in the segment of the rope that has the 10 kg mass
b.) What is the torque on the rope above the 5 kg mass?
c.) What is the angular acceleration?
d.) What is the velocity of the 10 kg mass after it has fallen 30 cm?

Calculate the magnitude of the electric potential difference across the length

9. A stainless steel tube having an outside diameter of 0.6 cm and a wall thickness of 0.05 cm is to be insulated with a material having a thermal conductivity of 0.065 W/mK. If the inner and outer convective heat transfer coefficients are 5.9 W/m²K. what will be the heat loss per meter of length for insulation thicknesses of 0, 0.25, 0.5, 0.75, 1.00, and 1.25 cm if the inside temperature is 95 C and the outside temperature is 10 C? Plot heat loss per meter versus insulation thickness. Also, plot the surface and interface temperatures as a function of insulation thickness.

a) DERIVE the equation for the change in the free energy of a CYLINDRICAL nucleus heterogeneously forming from an existing phase. The nucleus grows with its circular base on

the mold wall and the axis of the cylinder normal to the wall. ASSUME that the height of the cylinder can be written as h = a * r, where a is a constant, and r is the cylinder radius.

b) Find the critical radius and the energy barrier to nucleation for this shape of precipitate.

c) We can vary the shape of the nucleating phase by changing the constant a. Make a log- log plot of the size of the energy barrier versus the parameter a over a range from a

= 0.01 (disk shaped) to a = 100 (needle shaped). Explain your results.

Dark matter; C&O 24.22.

(a) Assume that the density of dark matter in our Galaxy is given by ρ(r) = ρ0/(1+(r/a) ^2) . Show that the amount of dark matter interior to a radius r is given by: Mr = 4πρ0a^2 [ r − a tan^−1 ( r/a )].

(b) If 5.4 × 10^11 M⊙ of dark matter is located within 50 kpc of the Galactic centre, determine ρ0 in units of M⊙ kpc−1 . Repeat your calculation if 1.9 × 10^12 M⊙ is located within 230 kpc of the Galactic centre. Assume that a = 2.8 kpc.

Suppose that you decided to send a spacecraft to Neptune using a Hohmann transfer. The craft starts in a circular orbit close to the Earth (1 AU) and is to end up in a circular orbit near   Neptune (about 30 AU).   (a) How long would the transfer take?   (b) How could you shorten this time, without increasing the amount of fuel required?   (c) What change in velocity is needed to enter the transfer orbit?   (d)   What is the change in velocity needed to enter the circular orbit near Neptune?