Math Is Fun Forum

Pratheepa

Member

Registered: 2007-12-01

Posts: 5

projectile

can some body help me with equations of projectile or trajectories????
wat r the equation to find the time,height,speed and other stuf?
i tried the wikipedia its confusing me...

Pratheepa

Member

Registered: 2007-12-01

Posts: 5

Re: projectile

can some one please help me with that question i asked about projectile.

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: projectile

Ill show you how they are all derived.

First of all, a projectile shooting off from ground level, on flat ground at angle θ to the ground, at u m/s.
Since we always ignore air resistance, the horizontal velocity is constant, and the vertical velocity is affected by gravity.

So as a bit of preliminary we can express the velocity at any point in time.

and thus the displacement at some time is

Time of flight

the time of flight can be found with vertical component of displacement

obviously the solution 0 isn't used, thats when the projectile took off from the ground

Height

The height of the trajectory will occur at a critical point in the vertical displacement, to find critical points we equate the dervitive to 0, in this case, the derivitave is the vertical velocity so we have:

(Notice this is half the time of flight, so this is another way of deriving time of flight)

Plug it into vertical displacement

Range

Range is found by plugging in the time of flight into the horizontal displacement, so we have:

Double angle formula:

so we have:

Maximum Range

To derive maximum range, differentiate equation for range with respect to angle to find maximum point, and plug back in.

You can just see that the maximum value of sin(2θ) = 1, which gives in the range, θ = 45, but its nice to explicitily derive it aswell ^^

anyways, plugging back into it

Maximum height

The maximum height reached with varying θ, using same method as range.

Again you could have simply looked at the formula, or even without imagined it, and gotten that value, but its nice to derive.

Equation of trajectory

Recalling from above the equation for displacement
let x be horizontal displacement, and y be vertical displacement we have:

We want to get rid of time from the equation, and have y expressed as a function of x, so rearrange x for t:

and plug it into y

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