# Motion in two and three dimensions question?

• Questions999
In summary, Galileo's theory states that if we ignore air resistance, the ranges of projectiles on a level field with angles of projection exceeding or falling short of 45 degrees by the same amount will be equal. This can be proven by finding the duration of the flight using the equations for horizontal and vertical speed, and then manipulating the formula for range to show the relationship between initial velocity and angle of projection.
Questions999
Galileo shows that, if any effects due to air resistance are ignored, the ranges for projectiles on a level field whose angles of projection exceed or fall short of 45 degrees by the same amount are equal. Prove this result.

A: So ,I tried these vx = v*cos(q) //q is the shooting angle, vx is the speed in horizontal direction
vy_0 = v*sin(q) //original vertical speed
vy = vy_0 - gt // the projectile is pulled down by the gravity
----
x(t) = vxt
y(t) = vy_0t - 0.5gt^2
----
when the projectile hits the ground, y is 0
vy_0t = 0.5gt^2
vy_0 = 0.5gt
t = 2vy_0/g
How do I relate these altogether to prove the Galileo thing? Please.

You have found the duration of the flight. Now find the horizontal distance covered in the flight; this is the range in the question. The range will depend on the initial velocity and the angle of projection, so manipulate its formula to show what is required.

## 1. What is motion in two and three dimensions?

Motion in two and three dimensions refers to the movement of an object in both the x-y plane and the z-axis. This means that the object is moving not only horizontally and vertically, but also has a component of motion in the depth or height direction.

## 2. What is the difference between motion in two dimensions and motion in three dimensions?

The main difference between motion in two and three dimensions is the number of axes involved. Motion in two dimensions only involves the x and y axes, while motion in three dimensions involves the x, y, and z axes. This means that in three dimensions, the object can move in three different directions, while in two dimensions, it can only move in two.

## 3. How is motion in two and three dimensions represented mathematically?

Motion in two and three dimensions can be represented using vector equations, which take into account the magnitude and direction of the object's motion. These equations can be broken down into components along each axis, usually denoted as x, y, and z, to describe the object's position, velocity, and acceleration in each direction.

## 4. What are some real-life examples of motion in two and three dimensions?

Examples of motion in two dimensions include a car driving on a flat road, a person walking on a flat surface, or a projectile motion like a ball being thrown through the air. Examples of motion in three dimensions include an airplane flying through the sky, a boat sailing on the ocean, or a rocket launching into space.

## 5. How is motion in two and three dimensions used in scientific research?

Motion in two and three dimensions is an important concept in many fields of science, including physics, engineering, and biology. It is used to study and analyze the movement of objects, understand the forces acting on them, and predict their future motion. This information can be applied to various real-world problems, such as designing efficient transportation systems, improving athletic performance, or developing medical treatments for movement disorders.

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