Projectile motion describes anything you throw, launch or fire that then travels under gravity alone. This projectile motion calculator takes your launch speed, angle, gravity and starting height, then works out how far the object travels, how high it rises, how long it stays in the air and the angle it comes down at, which are the four numbers that decide whether you hit your target.
What Is the Projectile Motion Calculator?
A projectile is any object that is launched and then moves only under the influence of gravity. The analysis splits the motion into two independent parts: the horizontal component, which travels at a constant speed because there is no horizontal force, and the vertical component, which accelerates downwards at the rate of gravity.
That separation is the whole trick. Once you split the launch velocity into its horizontal and vertical parts, every other result follows from the standard equations of accelerated motion.
How the Projectile Formulas Work
vx = v × cos(θ) and vy = v × sin(θ), with the launch angle θ measured from the horizontal.
The time of flight comes from the vertical motion, where the final vertical speed is zero at the peak, and the range is simply the horizontal speed multiplied by that time. The maximum height follows from the energy-style relationship between the initial vertical speed and the height gained.
The impact angle is found from the final velocity components, which is the figure that matters when you are trying to work out how steeply a ball will drop into a receiver or a bucket.
Choosing the Right Angle
On level ground the range is greatest at 45 degrees, and the calculator shows the maximum range at your chosen speed so you can see how much a different angle would cost you. That maximum range is simply the square of the launch speed divided by gravity.
If you need to clear an obstacle instead of maximise distance, use a higher angle. If you are launching into a strong crosswind, a flatter angle usually works better because a high arc gives the wind more time to push the object sideways.
What This Model Leaves Out
The calculation assumes no air resistance, which is a good approximation for heavy objects over short distances but becomes badly wrong for light objects over long ones. A golf ball travels about twice as far as this model predicts because of lift and drag.
It also assumes a flat landing surface. On sloping ground the range changes, and a projectile launched on an incline has a range of twice the range obtained on level ground with the same launch angle measured from the slope.
Frequently Asked Questions
What angle gives the maximum range?
Forty five degrees above the horizontal gives the greatest range when the launch and landing points are at the same height. The maximum range itself is the launch speed squared divided by the gravity.
Does air resistance change the answer?
Substantially for light or slow objects. A golf ball travels roughly twice the range this model predicts, and a thrown tennis ball loses most of its range over 20 metres. The calculator is reliable for dense objects thrown at moderate speeds over short distances.
How do I calculate the time of flight?
Solve the vertical motion for when the vertical position returns to the launch level. For a ground launch that is two times the vertical speed divided by gravity, and the calculator adds the launch height term when you start above the landing surface.
What does the impact angle tell me?
It is the angle the object is travelling at as it reaches the ground, measured below the horizontal. It matters for catching a ball, for the trajectory of a shell, and for judging how steeply a dropped object will enter a target.
Does the launch height change the range?
Yes. Launching from a height gives more time in the air, and therefore a longer range, even with the same speed and angle. The calculator includes the launch height in the time of flight calculation.