What is projectile motion? Projectile motion definition
Imagine an archer sending an arrow in the air. It starts moving up and forward, at some inclination to the ground. The further it flies, the slower its ascent is – and finally, it starts descending, moving now downwards and forwards and finally hitting the ground again. If you could trace its path, it would be a curve called a trajectory in the shape of a parabola. Any object moving in such a way is in projectile motion.
Only one force acts on a projectile – the gravity force. Air resistance is always omitted. If you drew a free body diagram of such an object, you would only have to draw one downward vector and denote it “gravity”. If there were any other forces acting on the body, then – by projectile motion definition – it wouldn't be a projectile.
Projectile motion analysis
Projectile motion is pretty logical. Let's assume you know the initial velocity of the object V, the angle of launch α, and initial height h. Our projectile motion calculator follows these steps to find all remaining parameters:
Calculate the components of velocity.
components of velocity
The horizontal velocity component Vx is equal to V * cos(α).
The vertical velocity component Vy is equal to V * sin(α).
Three vectors - V, Vx and Vy - form a right triangle.
If the vertical velocity component is equal to 0, then it's the case of horizontal projectile motion. If, additionally, α = 90° then it's the case of free fall.
Write down the equations of motion.
Horizontal distance traveled can be expressed as x = Vx * t, where t is the time.
Vertical distance from the ground is described by the formula y = h + Vy * t – g * t² / 2, where g is the gravity acceleration.
Horizontal velocity is equal to Vx.
Vertical velocity can be expressed as Vy – g * t.
Horizontal acceleration is equal to 0.
Vertical acceleration is equal to -g (because only gravity acts on the projectile).
Calculate the time of flight.
time of flight
Flight ends when the projectile hits the ground. We can say that it happens when the vertical distance from the ground is equal to 0. In the case where the initial height is 0, the formula can be written as: Vy * t – g * t² / 2 = 0. Then, from that equation, we find that the time of flight is
t = 2 * Vy / g =2 * V * sin(α) / g.
However, if we're throwing the object from some elevation, then the formula is not so nicely reduced as before, and we obtain a quadratic equation to solve: h + Vy * t – g * t² / 2 = 0. After solving this equation, we get:
t = [V * sin(α) + √((V * sin(α))² + 2 * g * h)] / g
Calculate the range of the projectile.
range
The range of the projectile is the total horizontal distance traveled during the flight time. Again, if we're launching the object from the ground (initial height = 0), then we can write the formula as R = Vx * t = Vx * 2 * Vy / g. It may be also transformed into the form: R = V² * sin(2α) / g
Things are getting more complicated for initial elevation differing from 0. Then, we need to substitute the long formula from the previous step as t:
R = Vx * t = V * cos(α) * [V * sin(α) + √(V * sin(α))² + 2 * g * h)] / g
Calculate the maximum height.
maxheight
When the projectile reaches the maximum height, it stops moving up and starts falling. It means that its vertical velocity component changes from positive to negative – in other words, it is equal to 0 for a brief moment at time t(Vy=0).
If Vy – g * t(Vy=0) = 0, then we can reformulate this equation to t(Vy=0) = Vy / g.
Now, we simply find the vertical distance from the ground at that time: hmax = Vy * t(vy=0) – g * (t(Vy=0))² / 2 = Vy² / (2 * g) = V² * sin(α)² / (2 * g)
Fortunately in the case of launching a projectile from some initial height h, we need to simply add that value into the final formula: hmax = h + V² * sin(α)² / (2 * g)
Projectile motion equations
Uff, that was a lot of calculations! Let's sum that up to form the most essential projectile motion equations:
Launching the object from the ground (initial height h = 0)
Horizontal velocity component: Vx = V * cos(α)
Vertical velocity component: Vy = V * sin(α)
Time of flight: t = 2 * Vy / g
Range of the projectile: R = 2 * Vx * Vy / g
Maximum height: hmax = Vy² / (2 * g)
Launching the object from some elevation (initial height h > 0)
Horizontal velocity component: Vx = V * cos(α)
Vertical velocity component: Vy = V * sin(α)
Time of flight: t = [Vy + √(Vy² + 2 * g * h)] / g
Range of the projectile: R = Vx * [Vy + √(Vy² + 2 * g * h)] / g
Maximum height: hmax = h + Vy² / (2 * g)
Using our projectile motion calculator will surely save you a lot of time. It can also work 'in reverse'. For example, enter the time of flight, distance, and initial height and watch it do all calculations for you!
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