Jonah Patel
Projectile motion is the motion of an object that is thrown or projected into the air and moves under the influence of gravity alone, with air resistance neglected. The object is called a projectile, and its curved path is known as its trajectory. The motion of falling objects is a simple one-dimensional type of projectile motion with no horizontal movement. Two-dimensional projectile motion, such as that of a football or basketball, can be analysed by breaking it into two independent one-dimensional motions along the vertical and horizontal axes. The key to this analysis is that motions along perpendicular axes are independent and can be analysed separately. The range of a projectile is the horizontal distance travelled on level ground. The greatest height that the object will reach is known as the peak of the object's motion.
| Characteristics | Values |
|---|---|
| Definition | Projectile motion is the motion of an object thrown or projected into the air. |
| Forces acting on the object | The only force acting on the object is gravity. |
| Air resistance | Air resistance is negligible. |
| Trajectory | The path through which the projectile travels is known as a trajectory. The trajectory may be parabolic or straight in the case of an object thrown directly upward or downward. |
| Velocity | The horizontal and vertical velocities are independent of each other. The horizontal velocity is constant, while the vertical velocity decreases as the object rises and increases as it falls. |
| Acceleration | Acceleration is uniform in the vertical direction and zero in the horizontal direction. |
| Range | The range is the horizontal distance traveled by a projectile on level ground. The range does not depend on the mass of the projectile. |
| Maximum height | The maximum height is the highest vertical position along the trajectory. The maximum height depends on the initial velocity and launch angle. |
| Time of flight | Time of flight is the time taken by the projectile to travel from the point of projection to the horizontal range. |
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What You'll Learn
Projectile motion
The motion of falling objects is a simple one-dimensional type of projectile motion in which there is no horizontal movement. In this case, the object is projected vertically upward, and the only force acting on it is gravity, pulling it downward. Two-dimensional projectile motion, on the other hand, involves independent motions along perpendicular axes, namely the horizontal and vertical axes. The horizontal motion occurs at a constant velocity, while the vertical motion experiences uniform acceleration. The x- and y-motions can be recombined to give the total velocity at any given point on the trajectory.
To analyse two-dimensional projectile motion, it is crucial to establish a coordinate system with an origin for the x and y positions. Typically, the initial position of the object is chosen as the origin, with x0 = 0 and y0 = 0. The positive vertical direction is usually defined as upward, while the positive horizontal direction is often associated with the direction of the object's motion. However, it is important to note that the coordinate system can be defined differently based on specific scenarios.
The range of a projectile on level ground refers to the horizontal distance it travels. The range equation assumes that the Earth's surface is flat, and any combination of trajectories that add up to 90 degrees will result in the same range. If the initial speed is high enough, the projectile can achieve escape velocity and go into orbit.
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Horizontal and vertical velocity
Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. The path of the object is called its trajectory. When an object is thrown into the air, its motion can be analysed by breaking it down into two independent one-dimensional motions along the vertical and horizontal axes.
The horizontal velocity of a projectile remains constant in the absence of air resistance. This means that the horizontal velocity of a projectile is the same as its initial horizontal velocity. This is because, in the horizontal direction, there is no acceleration to change its speed. Therefore, the horizontal velocity of a projectile is independent of the influence of gravity.
The vertical velocity of a projectile, on the other hand, is subject to the influence of gravity. As the object rises, its vertical velocity decreases until it reaches its highest point, at which its vertical velocity is zero. As the object falls back towards the Earth, its vertical velocity increases in magnitude but points in the opposite direction to its initial vertical velocity. The vertical velocity of a projectile can be calculated based on gravity and time of flight.
The interplay between the horizontal and vertical velocities determines the overall trajectory and impact velocity of a projectile. For example, increasing the angle of projection increases the time of flight, leading to a change in vertical velocity due to additional time under gravity's influence. At the same time, an increase in initial velocity boosts both horizontal and vertical velocities, altering the overall trajectory and impact velocity of the projectile.
Understanding the horizontal and vertical velocities of a projectile is crucial in various practical applications, such as safety assessments, sports strategies, and space research. For instance, in a high dive competition, divers catapult themselves vertically upward off the board while also adding some horizontal velocity to reach the pool centre. Similarly, in sports like football, when a player kicks the ball at a certain force and angle, the ball follows a curved (parabolic) path due to the vertical velocity bearing the brunt of gravity, while the horizontal velocity remains constant.
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Air resistance
The drag force exerted by air resistance is always in the opposite direction to the velocity of the object. This force is influenced by several factors, including the density of the fluid (in this case, air), the cross-sectional area of the object, its speed, and the dynamic viscosity of the fluid. The magnitude of the drag force is described by the dimensionless drag coefficient, which is calculated using these variables.
In the context of projectiles, air resistance plays a crucial role in determining their range and trajectory. When a projectile is launched, there is an initial time interval during which the trajectory is unaffected by air resistance. However, as air resistance strength increases, the range of the projectile decreases, and the projectile tends to fall more steeply than it rises. In the presence of strong air resistance, the projectile may even fall almost vertically.
The impact of air resistance on the horizontal range of a projectile is significant. It causes the horizontal range to scale linearly with the launch velocity, rather than quadratically, as observed in the absence of air resistance. Additionally, the maximum horizontal range is achieved with a shallower launch angle when air resistance is considered.
It is important to note that in introductory physics and dynamics courses, gravity is often the only force considered, neglecting the effects of air resistance. However, realistic and accurate models of air resistance are essential for understanding the motion of projectiles in the real world, such as the trajectory of a baseball or a cannonball in flight.
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Trajectory
The trajectory of a projectile is the path it follows when launched or thrown into the air. The trajectory of a projectile is parabolic, though the path may also be straight if the object is thrown directly upward or downward.
In physics, projectile motion describes the motion of an object that is launched into the air and moves under the influence of gravity alone, with air resistance neglected. This is a simplified model, as in reality, air resistance is a frictional force that slows the motion of a projectile and can significantly alter its trajectory. However, when the deviation from projectile motion is negligible, air resistance can be ignored.
Projectile motion can be decomposed into horizontal and vertical components. The horizontal motion occurs at a constant velocity, while the vertical motion experiences uniform acceleration due to gravity. These motions are independent of each other, meaning neither motion affects the other. This principle of compound motion was established by Galileo in 1638, who used it to prove the parabolic form of projectile motion.
The range of a projectile is the horizontal distance it travels on level ground. The range does not depend on the mass of the projectile but on its velocity and direction. The maximum range is obtained when the projectile is launched at an angle of 45 degrees.
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Acceleration
The trajectory of a projectile can be analysed by breaking its motion into two independent one-dimensional motions: horizontal and vertical. The horizontal motion of a projectile occurs at a constant velocity, meaning it does not accelerate in the horizontal direction. However, the vertical motion experiences uniform acceleration due to the force of gravity. As the object rises, its vertical velocity decreases until it reaches its highest point, where the vertical velocity momentarily becomes zero. As the object falls back towards the Earth, its vertical velocity increases, but in the opposite direction to its initial upward velocity.
The total velocity of the projectile at any given point in its trajectory can be determined by combining its horizontal and vertical motions. The horizontal velocity remains constant, while the vertical velocity changes as the object rises and falls, resulting in a curved path known as a parabola. This parabolic trajectory was first demonstrated by Galileo Galilei in the 17th century and forms the basis of classical mechanics.
The acceleration of gravity also changes with the altitude of the projectile. As the projectile gains altitude, the acceleration due to gravity decreases in magnitude. This change in acceleration can result in an elliptic trajectory, especially when considering the curvature of the Earth. If the initial speed of the projectile is high enough, it can even escape the Earth's gravitational pull and go into orbit.
The study of projectile motion and acceleration has numerous practical applications, including ballistics, engineering, sports science, and the understanding of natural phenomena. By considering the initial velocity, launch angle, and acceleration due to gravity, we can predict the trajectory, maximum height, and time of flight of a projectile.
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Frequently asked questions
A projectile is any object that is thrown or projected into the air and experiences a force of acceleration due to gravity.
Projectile motion is the movement of a projectile, which can be broken down into two independent one-dimensional motions: horizontal and vertical.
Horizontal motion occurs at a constant velocity, while vertical motion experiences uniform acceleration due to gravity.
The range of a projectile is influenced by its initial velocity and launch angle. The greater the initial speed and the optimal launch angle, the greater the range.
Air resistance is a frictional force that acts on a projectile, slowing it down and altering its trajectory. In some cases, air resistance can be neglected in calculations if it is minimal.
