The Proof is Trivial physics edition?

I have no idea what M1 is but yeah sorry your solution is correct

M1 is the first mechanics module in A level maths. This could have been copied out of my textbook

Explain why a glass prism can separate white light.

I wrote this my self in first year for an online thing we had to do where you make questions and everyone answers and rates it. You wouldn't believe how many people got it wrong haha

(..)

A sphere of mass 3 kg is rolling along a surface (without slipping). It moves with a constant velocity of 2 ms^-1 until it reaches a ledge and lands in a fluid where all of its energy is converted into rotational energy and it stays spinning in the fluid. Given the radius of the sphere is 7 cm, calculate the angular velocity of the sphere after it lands in the fluid and which direction the angular velocity vector points (into screen/out of screen)

A sphere of mass 3 kg is rolling along a surface (without slipping). It moves with a constant velocity of 2 ms^-1 until it reaches a ledge and lands in a fluid where all of its energy is converted into rotational energy and it stays spinning in the fluid. Given the radius of the sphere is 7 cm, calculate the angular velocity of the sphere after it lands in the fluid and which direction the angular velocity vector points (into screen/out of screen)

Is it cheating to look up the kinetic energy of a rotating sphere? Because I do not know it/a sphere's moment of inertia off the top of my head and my calculus isn't too good.

Not sure about direction. (Don't post a hint though!)

Is the height of the drop negligible?

I think the idea of the force acting as it hits the surface is a bit ambiguous, there is a lot more considerations to be made. But you could just ignore it for this question, as all the energy is transferred into rotational energy, and we can't make the assumption it'll rotate in the other direction. The direction though is correct.

When a proton, p, collides at sufficiently high energy with another proton, a proton anti-proton pair can be created in addition to the original protons.

a). Give the equation for this interaction

b). How much addition energy is required to create these particles?

Assume the force is always to the left and is to some function so that it always brings the translational kinetic energy to zero- for all the time that the ball is up to half-submerged, the force causes a clockwise rotation because it acts only on the bottom half. Once the ball is fully submerged, the fluid exerts a force to the left over the entire surface of the sphere and so doesn't cause any rotation. Although this makes some assumptions about the nature of the force, I think this is a reasonable analysis.

Moment of inertia of a sphere rotating around its centre is 2/5mr^2

a) p + p --> p + p + p + p(bar)
b) 1876 MeV

then:
$1/2 I \omega ^2 = 1/2 I \omega_0^2 +1/2mv^2[br]2/5 mr^2 \omega^2=2/5mr^2\omega_0^2+mv^2$

Since the ball travels a distane 2*pi*r for every 2*pi rotation origionally, we can write:
$v=\omega_0 r[br]2/5 r^2 \omega^2 = 2/5 r^2 \omega_0^2 + \omega_0^2 r^2[br]\omega ^2=7/2 \omega_0^2 = \dfrac{7v^2}{2r^2}[br]$

After actually substituting values I get: 53 rad/sec? that seems too fast...

Explain why a glass prism can separate white light.

Yep that's what I got If you consider that the final answer doesn't depend the mass, only on the inital velocity and the radius of the ball. Since the ball is travelling at 2 m/s when the radius is only 0.07 cm then it spins at 4.55 revs/sec.
In the fluid it is 53 rad/sec which is 8.43 revs/sec so it isn't that much of an increase really

Okay, I couldn't see what could have gone wrong since the thing making it that high was simply the radius being so small. Good question!