- 11 Aug '11 00:30This is from an Open University website

http://www.open.ac.uk/openlearn/whats-on/ou-on-the-bbc-the-code-kingdom-catapults-hints-and-tips

I object to one of these statements.

Can any of you spot which one?

And do you agree with me that it is wrong?

I have mailed the OU with my objection and I will post any response they give if and when I receive it.

"(1) Kingdom of Catapults is related to the mathematical theory of projectiles.

This theory is concerned with the motion of an object that is launched into the air.

(2) In Kingdom of Catapults you have control of the launch angle and launch speed of projectile pieces of fruit. Altering the speed and angle affect both the path of the projectile, and the time it takes for the projectile to land. The fruits are of different weights, so some can be launched with greater initial speeds than others.

(3) We ignore air resistance in Kingdom of Catapults, which means that the initial launch angle and speed determine the motion of the projectile completely. The distance travelled by the projectile can be calculated from an equation.

Mathematical methods and models

(4) The path followed by a projectile is known as its trajectory. Without air resistance the trajectory has the shape of a parabola. A parabola is a mathematical curve with many important properties. For example, the mirrors in car headlights have a parabolic shape in order to focus light in a beam.

(5) Remarkably, without air resistance, the trajectory and time of flight of a projectile do not depend on the mass of the projectile. A cannon ball will travel the same distance and for the same length of time as a marble. In reality, a cannon ball would feel more air resistance (and the marble would travel farther, provided that it had the same density as the cannon ball).

(6) The trajectory of a projectile is also affected by gravity. Because the Sun has greater mass than the Earth, a projectile fired from the Earth would travel farther than the same projectile fired from the Sun. On the other hand, a projectile fired from the Moon would travel farther than the same projectile fired from the Earth.

(7) When you fire a projectile vertically in the air the projectile is, for an instant, motionless when it reaches its highest point. The projectile is never motionless when the launch angle is not vertical; however, at the highest point of its parabolic trajectory its vertical speed is zero, but its horizontal speed is not zero.

(8) In fact, without air resistance there are no horizontal forces acting on the trajectory, and its horizontal speed remains constant throughout its motion.

(9) When firing from flat ground, and ignoring air resistance, the maximum range of a projectile is achieved when you fire at a 45 degree angle to the horizontal ground. In Kingdom of Catapults you fire from a high castle, and the maximum range is achieved when you fire at an angle less than 45 degrees to the horizontal. (The precise angle depends on the height of the castle.)

(10) The equations of motion of a projectile are more complicated when air resistance is considered. The size and shape of the projectile both affect air resistance. The trajectory is no longer typically a parabola." - 11 Aug '11 12:44What they don't talk about is the force required to fire the projectiles. There seems to be the assumption that the initial velocity will be the same regardless of the objects mass.

The cannon ball experiences greater air resistance but has less momentum. To be honest, I don't know which would go further. - 11 Aug '11 13:57It's not what they mean, I suspect, but switch "density" to "mass" then it'll be accurate.

The air resistance is, roughly speaking, proportional to the cross-sectional area. So for objects of the same shape and density, the effect of air resistance will be greater on the smaller object.

(I think at that point they're talking about having the same initial velocity - the context is talking about independence from mass without air - that's only true given the same initial velocity). - 11 Aug '11 16:47Drag is proportional to area which is a function of the square of the size.

Mass is proportional to volume which is a function of the cube of the size.

Thus for objects with equal densities and identical shapes with the same initial

velocity the larger object will experience less force per unit mass

(i.e. less acceleration) due to drag.

Thus the smaller object will decelerate faster and thus go the shorter distance.