Trajectory Formula with Problem Solution & Solved Example


What is Trajectory?

Trajectory is defined as the path preferred by an object through the space as the function of time. In mathematical terms, it could also be defined as the position of an object within a specific time interval. To under the concept in a better way, you could also consider the example of a ball that is thrown upwards and path taken by the ball against the gravitational force or air resistance.

Trajectory Formula

Trajectory Formula

One more example could be considered here for bullet fired from the gun or satellite moving toward the Earth orbit. The trajectory formula in mathematics is written as below –

\[\large y=x\:tan\,\theta-\frac{gx^{2}}{2v^{2}\,cos^{2}\,\theta}\]

Where,
y is the horizontal component,
x is the vertical component,
g= gravity value,
v= initial velocity,
θ = angle of inclination of the initial velocity from horizontal axis,

Trajectory related equations are:

\[\large Time\;of\;Flight: t=\frac{2v_{0}\,sin\,\theta}{g}\]

\[\large Maximum\;height\;reached: H=\frac{_{0}^{2}\,sin^{2}\,\theta}{2g}\]

\[\large Horizontal\;Range: R=\frac{V_{0}^{2}\,sin\,2\,\theta}{g}\]

Where,
Vo is the initial Velocity,
sinθ is the y-axis vertical component,
cosθ is the x-axis horizontal component.

The next associated term here is ballistic trajectory here that explain about the path taken by an object when it is initially launched in the space then its path could be defined through law of motion, air resistance, and the gravity. You must be wondering how to calculate the ballistic trajectory path here. Don’t get confused because this is easy with right formula and technique. However, you have to work on certain assumptions too still it is quite common in use and taken important part of higher studies in maths and science. When any object is thrown in a horizontal or vertical direction, it will be keep moving through a constant speed against gravity and air resistance. There are different equations for the horizontal or vertical trajectory movements still easy when used wisely.


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