Projectiles
Projectiles are objects propelled through the air by an initial force, after which their motion is largely influenced by gravity. This category includes a wide range of entities, such as sports balls, bullets, and even microscopic particles in a vacuum. Once a projectile is in motion, it follows a path known as a ballistic trajectory, which is determined by its initial velocity and the angle at which it was projected. The path typically resembles a parabolic shape, with its highest point referred to as the vertex, where vertical velocity momentarily reaches zero before descending.
A fundamental aspect of projectile motion is that horizontal and vertical movements operate independently; while a projectile continues to move horizontally at a constant speed due to inertia, its vertical speed decreases under the influence of gravity until it reaches zero at the vertex. The time a projectile spends in the air—its flight time—is governed solely by gravitational acceleration. Real-world factors like air resistance can affect the trajectory, causing deviations from the ideal parabolic path and reducing horizontal velocity over time. Understanding these principles is crucial for anyone engaging in activities that involve projectile motion, such as sports or engineering applications.
Projectiles
FIELDS OF STUDY: Classical Mechanics
ABSTRACT: Physical objects moving by inertia within a gravitational field are called projectiles. The study of their motion is called ballistics. The mathematics that describes ballistic motion applies to objects as small as a subatomic particle and as large as an intercontinental missile. The physical size of the projectile makes no difference. This enables the use of ballistics in many applications of physical science.
Principal Terms
- acceleration: the rate at which the velocity of an object increases over time.
- ballistic trajectory: the motion described by a projectile traveling by inertia in a gravitational field.
- deceleration: the rate at which the velocity of an object decreases over time.
- inertia: the resistance of an object to changes in its velocity.
- initial velocity: the velocity of an object at the start of some interval of time.
- parabolic: refers to a shape (a parabola) that can be described by an equation of the form y = ax2 + b.
- range: the horizontal distance that a projectile can travel before striking the ground.
- terminal velocity: the velocity at which the acceleration due to gravity of an object falling freely through a fluid medium, such as air or water, is exactly balanced by the deceleration of the object due to resistance from that medium.
- vertex: the uppermost point in a ballistic trajectory.
What Is a Projectile?
A projectile is any object that is shot through the air by an initial force. After this initial force, the projectile’s motion is affected only by the force of gravity. Examples of projectiles are numerous, including sand and stones thrown up by the spinning wheels of a car, a baseball struck by a bat, a bullet fired from a gun, and a satellite launched into orbit. Even very small particles such as ions (electrically charged atoms) traveling through a vacuum chamber are considered projectiles.
After a projectile has been given an initial velocity by some outside force, it maintains its horizontal velocity due to inertia. After that, it only changes its vertical position and speed because of gravity. When the initial force stops acting on the object, such as when a football leaves the hand of the person throwing it, the object becomes a projectile. The subsequent path it follows is called a ballistic trajectory. That trajectory is determined by the angle and velocity of the object when it first becomes a projectile. It is also determined by the force of gravity acting on the object.
The ballistic trajectory of a projectile is parabolic in shape. Its motion has only two directions, vertical and horizontal, which operate independently of one another. Generally, a projectile will move in both these directions, although it may have no horizontal movement at all. For example, a ball that is thrown straight up in the air and comes down in exactly the same spot has moved vertically but not horizontally.
Ballistics
A normal parabola is a symmetrical curve. Its shape is described by the mathematical formula y = ax2 + bx + c, where y represents the vertical displacement, x represents the horizontal displacement, and a represents the change in vertical position relative to horizontal position. The coefficients b and c simply change the position of the curve. The simplest form of this equation is y = ax2, where b and c both equal 0.
A ballistic trajectory differs somewhat from a normal parabola. Technically, a parabola is a curve whose ends go on forever. A ballistic trajectory, however, has distinct start and end points, separated by a horizontal range. The highest point of a ballistic trajectory is called the vertex. This is the point at which the projectile’s vertical velocity is zero, as it transitions from traveling upward to falling back down. Ideally, if the start point and the end point are at the same elevation, the vertex will be exactly halfway between the two.
The vertical velocity and the horizontal velocity of a projectile can be calculated separately, as neither affects the other. In an idealized model, as the projectile travels, its horizontal velocity will remain constant, while its vertical velocity will steadily decrease as it travels upward. This deceleration is due to the force of gravity acting on the projectile. When the vertical velocity reaches zero, the projectile can go no higher. It then begins to fall back to the ground, accelerating at the same rate that it decelerated previously. This rate is the acceleration due to gravity, which on Earth is approximately 9.81 meters per second per second (m/s2).
However, this model of ballistic trajectory does not take into account the medium that the projectile travels through. The medium creates resistance as objects pass through it, and the greater the density of the medium, the greater the resistance. A bullet from a high-powered rifle, for example, can travel several hundred meters through the air without losing much horizontal velocity. However, the same bullet fired into water will lose almost all of its horizontal velocity after just two or three meters.
A parachutist jumping from an airplane experiences the same thing as he or she falls through the air. When the parachutist exits the airplane, the initial horizontal velocity will be the same as that of the airplane, and the initial vertical velocity will be zero. Air resistance will cause the parachutist’s horizontal velocity to decrease until it is essentially zero and the parachutist is falling straight down. Meanwhile, the parachutist’s vertical velocity will increase at a rate of 9.81 m/s2 until the force of the air resistance is equal to the force of gravitational acceleration, at which point he or she can fall no faster. The speed at which this occurs is called terminal velocity and is typically reached long before the parachutist will deploy his or her parachute.
Practical Ballistics
The phenomenon of medium resistance means that in reality, the ballistic trajectory of a projectile will not have a perfectly parabolic shape. Resistance will cause the projectile’s horizontal velocity to decrease slightly over time, reducing its range. Also, the vertex will not be exactly halfway between the start point and the end point. It will be closer to the end point, because the decrease in horizontal velocity means that the projectile will travel a shorter horizontal distance after passing its vertex. This is a very important consideration for pilots, parachutists, target shooters, baseball players, basketball players, and anyone else who relies on ballistic motion to achieve the goal of hitting a target with a projectile.
Another important consideration is the flight time of a projectile following a ballistic trajectory. Because the horizontal velocity is not affected by the force of gravity, the projectile will continue to travel horizontally until the force of gravity has brought it into contact with the ground. This means that if a ball is thrown straight ahead in such a way that its initial vertical velocity is zero, it will travel through the air for the same amount of time as if it had been simply dropped from the same height. This is because the ball is accelerating toward the ground at the same rate in both cases. This length of time is the flight time, and it is determined solely by the force of gravity acting on the projectile. The greater the horizontal velocity of the projectile, the farther it will travel during the flight time. A slow projectile and a fast projectile will both strike the ground at the same time, however, as long as they are both thrown from the same point at a zero-degree angle.
Sample Problem
A person standing at the top of a 30-meter-high cliff throws a rock straight out at a zero-degree angle. The rock is released from a point 2 meters above the edge of the cliff and travels with an initial horizontal velocity of 25 meters per second. Ignoring the effects of air resistance, how far will the rock travel horizontally before striking the ground 32 meters below? Recall that the acceleration due to gravity is 9.81 meters per second per second.
Answer:
The simplest way to solve this problem is to calculate the amount of time it would take for the rock to fall straight down from a height of 32 meters, then calculate how far the rock will travel horizontally during that time at a velocity of 25 meters per second. If air resistance is ignored, it can be assumed that the horizontal velocity will remain constant throughout the flight time.
The initial vertical velocity of the rock is 0 m/s. Its final vertical velocity of the rock is not known, but it is known that the rock travels a distance of 32 m in free fall, accelerating at a rate of 9.81 m/s2. The final velocity of an object in free fall can be calculated using the following equations, known as equations of motion:
vf2 = v02 + 2as
vf = v0 + at
where v0 is the initial velocity, vf is the final velocity, a is acceleration, s is displacement (in this case, vertical displacement), and t is time. The variable v0 can be eliminated, because the initial velocity is 0. Therefore,
vf2 = 2as
vf = at
Solve the first equation for vf, combine both equations into one, and solve for t:
Multiply t (2.554 s) by the horizontal velocity (25 m/s) to determine the horizontal distance traveled:
2.554 × 25 = 63.85
The rock traveled a horizontal distance of about 63.85 meters.
Equations of Motion
Like all of physics, the description of ballistic trajectories relies heavily on mathematics. It is a fairly simple exercise to watch a projectile travel through the air and relate its motion to mathematics. This can be as simple as watching a stream of water come from a garden hose or tossing a ball into the air. The equations that describe the movements of objects through a constant gravitational field are called "kinematic equations" or "equations of motion." They can be remembered using the acronym VAST, which stands for velocity (v), acceleration (a), displacement (s), and time (t). As demonstrated in the sample problem, these equations can be rearranged and combined in such a way that if just two of these quantities are known, they can be used to calculate the other two.
Bibliography
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Grissom, Thomas. The Physicist’s World: The Story of Motion and the Limits of Knowledge. Baltimore: Johns Hopkins UP, 2011. Print.
Hamilton, Sue L. Forensic Ballistics: Styles of Projectiles. Edina: ABDO, 2008. Print.
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Stahler, Wendy, Dustin Clingman, and Kaveh Kahrizi. Beginning Math and Physics for Game Programmers. Indianapolis: New Riders, 2004. Print.
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