# Projectile Range Calculator

With a projectile range calculator, you can quickly find out how far the object can be thrown for a certain range. All you need to do is enter the three parameters of projectile motion (which are explained in the article as well) - velocity, angle, and height from which the projectile is launched. In the fractions of seconds, you will find the horizontal displacement of your object.

### What is meant by Projectile?

A projectile is an object in space or an object on earth with a standard mass propelled by the exertion of a force that is then allowed to move freely under the influence of gravity and air resistance. An object or a missile that once projected or dropped continues in motion by its own inertia and is influenced only by the downward force of gravity.

Hence by the definition, a projectile has a single force that acts upon it - the force of gravity. Although many objects in motion through space are projectiles, they are commonly found in warfare and sports. In physics, we generally tend to overrule the concept of air resistance to make the calculations easy.

### Range in Projectile Motion Formula

Let us split the equations into two cases for easy and better understanding: when we launch the projectile from the ground, that is the net height is taken to be zero, and when the object is thrown from some initial height (for example. table, building, bridge, inclined place, etc.). Air resistance of all directions is neglected in all calculations because as the matter of fact the air resistance is generally negligible for any object on the earth.

**CASE 1: Launch of the object under the influence of external force from the ground (here, the initial height is taken as 0)**

To find the formula for the range of such a projectile or the object, let us start from the basic equation of motion. The projectile range is the distance traveled by the object when it returns to the ground (so y, the horizontal component=0) 0 = V₀ * t * sin(α) - g * t² / 2

- Here V₀ is the vertical velocity of the object under influence of external force.
- T is the time taken for the projectile motion to complete its trajectory.
- Alpha (α) is the angle at which the object or the projectile is thrown in the air.
- g is the gravity of the place in which the experiment is being done(for most of them it would be earth, hence the value of ‘g’ can be taken as 9.8 m/s/s)

From this equation, we will find t, which is the time of flight to reach the ground t = 2 * V₀ * sin(α) / g

Also, we know that the maximum distance of the projectile can be found from the simple relation of the motion in a straight path defined by d = V * t. Velocity in our case is the horizontal velocity Vx = V0 * cos(α), and time to reach the ground is a value we've already calculated is as follow

d = V * t = V₀ * cos(α) * 2 * V₀ * sin(α) / g

d = 2 * V₀² * cos(α) * sin(α) / g

Knowing the trigonometric identity of sin(2 * x) = 2 * sin(x) * cos(x), we can rewrite the final formula as d = V₀² * sin(2 * α) / g. And hence this formula can be used as the standard formula for determining the range of the object under the influence of external force from a net height of zero.

**Case 2. Launch from an elevation (in this case the initial height > 0)**

In this particular case, the time spent flying upwards is much shorter than the time when the object is falling down in the same trajectory (time from reaching the maximum height to striking the ground in the same projectile or trajectory). The formula for the projectile range hence may be written as d = V₀ * cos(α) * [V₀ * sin(α) + √((V₀ * sin(α))² + 2 * g * h)] / g

- V₀ is the vertical velocity of the object under influence of external force.
- T is the time taken for the projectile motion to complete its trajectory.
- Alpha (α) is the angle at which the object or the projectile is thrown in the air.
- g is the gravity of the place in which the experiment is being done(for most of them it would be earth, hence the value of ‘g’ can be taken as 9.8 m/s/s)
- h is the net height from which the object or the projectile has been thrown.
- sin(α): stands for the vertical component of the angle.
- And cos(α)stands for the horizontal component of the angle. The net of both cos and sin is taken as the final angle in the calculations.

### Maximum Range and Range in a Projectile Motion

With the help of formulas, one can calculate at which angle an object or projectile be thrown so that it gains maximum height. We can solve this quite easily by differentiating the formula and equating it with zero hence getting the answer as 45 degrees. So 45 degrees is the only or the best angle at which the maximum height of the projectile can be achieved for any object or projectile following a trajectory under projectile motion.

The range of the projectile is defined as the displacement in the horizontal direction of the motion of the object. There is no acceleration in this direction of the motion since gravity only acts vertically on the object. This line is represented as a straight horizontal line in the diagrams. Like the time of flight and maximum height, the range of the projectile is a function and is linearly dependent on the initial speed of the object under the influence of the external force. The range of the projectile also depends on the angle at which the object is thrown.

For more concepts check out physicscalculatorpro.com to get quick answers by using this free tool.

### How does the Projectile Range Calculator work?

The projectile range calculator which we have developed for you is helpful to reduce your time and energy and get an instant answer to any question which demands the range in the projectile motion or the trajectory, as soon as possible. Here is how it works, though the basic operation on the usage of the formulas is what it operates on.

- The projectile range calculator asks the user to input the required parameters like velocity, angle of launch, initial height, range, etc.
- After getting all the parameters (that are height, vertical velocity ‘v’, horizontal velocity ‘u’, Angle and time) as inputs, the calculator works with the formula of projectile motion and trajectories depending on a particular case to give a solution.
- Finally after tapping on the calculate button the range of the projectile is displayed as the output.

### Projectile Motion Range Examples

**Question 1:** An object in the influence of an external force is launched at a velocity of 40 m/s in a direction making an angle of 40° upward with the horizontal. Calculate the range that is covered by the object? Take the time of flight as 6.25 sec to make the calculations easier.

**Solution:**

Given: The time of flight - 6.25 sec

The velocity of an object in ‘y’ direction as 40m/s

The angle of the projection of the object is 40 degrees

The Horizontal Range is the horizontal distance given by x at t = t2.

Hence by the formula for calculating range: R= d = V₀² * sin(2 * α) / g

Substituting the values in the above formula we get,

R (range)= 6.25 metres (all the calculation is done in standard units)

**Question 2:** Let us suppose a cricket player hit a ball in the match, striking it away from the bat at a velocity of 45.0 m/s at an angle of 66.4 degrees in relation to the field and the ground. Moreover, if the direction of travel of the ball is towards the end of the field which is 140.0 m away from the batsman. Then what will be the height of the ball when it will reach the end of the field? What is the maximum range of the ball?

**Solution:**

Given :

The velocity at which the ball is struck from the ground is 45 m/s

The angle at which the ball is struck from the ground is 66.4 degree

Horizontal direction (in meters) as covered by the ball is 140 m

Hence to calculate the range of the ball (r) = 2 * V₀² * cos(α) * sin(α) / g

Substituting the values in the given formula: we get 24.2 meters.

Hence the range of the ball at the edge of the ground is 24.2 meters.(all the calculation is done in standard units)

### FAQs on Projectile Range Calculator

**1. How do you calculate the range of a projectile?**

Range of Projectile can be found using the formula d = V₀² * sin(2 * α) / g, where v stands for velocity, alpha for the angle at which the projectile has been thrown, and g for the gravity of the place in which the experiment is being done.

**2. What is meant by the range of the projectile?**

The range of the projectile is defined as the maximum horizontal distance that can be covered by an object under the external force.

**3. What is vertical velocity?**

Velocity in the vertical motion of the object or in the upward/downward motion of an object is known as vertical velocity.

**4. What is a horizontal velocity?**

Velocity in the horizontal motion of the object or the projectile thrown is nothing but the horizontal velocity.

**5. What is projectile motion?**

Any object that we throw in the air and cover a path of trajectory is said to be in projectile motion.