9.2.B - Projectiles


Gravity

Projectile motion is one of the traditional branches of classical mechanics. A projectile is any body that is given an initial velocity and then follows a path determined by the effect of gravitational acceleration. The path of the projectile (trajectory) is determined by its initial velocity (magnitude and direction) and the acceleration due to gravity. Kicked footballs, objects dropped from airplanes and bullets shot from a gun are all examples of projectiles.

Projectile motion is usually analysed by considering the horizontal and vertical components separately. There is no acceleration in the horizontal dimension; however, gravitational acceleration acts in the vertical dimension.

Projectile motion may only be used to solve mechanics problems if the acceleration is constant.

In this unit you will learn about:

  • Galileo's analysis of projectile motion
  • How to describe the path and motion of a trajectory
  • How to analyse and calculate aspects of the motion of a trajectory such as time of flight, velocity and range


Galileo's Analysis of Projectile Motion

The general opinion nefore Gallileo.

Gallileo's sketches of projectile motion.

Before Galileo, people assumed that an object shot from a cannon would follow a straight line until it "lost its impetus," at which point it fell abruptly to the ground. This understanding is shown in the first illustration on the right.

Later, simply by more careful observation, it was realised that projectiles actually follow a curved path. Yet no one knew what that path was, until Galileo. Here was yet another brilliant insight that led Galileo to his most astounding conclusion about projectile motion. He reasoned that two types of motion, rather than one, influence the trajectory of a projectile. The force of gravity influences the motion that acts vertically and this pulls an object towards the Earth at 9.8 ms-2. But while gravity is pulling the object down, the projectile is simultaneously moving forward horizontally at a constant velocity.

Galileo was indeed able to show that two independent motions, horizontal and vertical, control the trajectory of a projectile and that they work together to create a perfect parabolic curve.

Projectile Motion

Vertical Motion

Both balls fall at the same
rate regardless of their
masses.

At a given location on the Earth and in the absence of air resistance, all objects fall with the same uniform acceleration. It follows that two objects of different sizes and weights, dropped from the same height, will hit the ground at the same time.

The vertical component of projectile motion is motion that is being accelerated downwards at 9.81 ms-2.

For this reason, the equations of motion can be used when working with the vertical component of a projectile's motion. These equations are used to determine the final vertical velocity, vy, initial vertical velocity, uy, vertical displacement, sy, and time of flight, t.

 

 

Equations - Calculating Vertical Projectile Motion Components

v

u

a

t

s

final vertical velocity

initial vertical velocity (uy = vsinθ)

acceleration due to gravity (g)

time of flight

vertical displacement (height)

metres per second (ms-1)

metres per second (ms-1)

metres per second squared (ms-2)

seconds (s)

metres (m)

You should use this equation when you are calculating vertical components of a projectile's motion.

 

Horizontal Motion

Each ball has the same mass
but the one on the right is
moving both vertically down
and horizontally.

The path of a projectile can be analysed in terms of two independent motions. So when an object is projected horizontally, it will reach the ground in the same time as an object dropped vertically. It does not matter how large the horizontal velocity is as the downward pull of gravity is always the same. There is no acceleration in the horizontal direction so the only equation that applies when working with horizontal motion is that which to calculate velocity when there is no acceleration.

The horizontal motion of a projectile is constant (velocity) as there is no net force in this direction to provide any acceleration.

The horizontal displacement of a projectile is also called the range. In most circumstances, you will need to determine the time of flight from the vertical motion before you can calculate the range.

 

Equations - Calculating Horizontal Projectile Motion Components

ux

s

t

initial (and final) vertical velocity (ux = vcosθ)

horizontal displacement (range)

time of flight (from vertical motion)

metres per second (ms-1)

metres (m)

seconds (s)

You should use this equation when you are calculating the range or horizontal velocity of a projectile.

 

Combining Motions

The illustration below shows how the accelerated vertical motion and the non-accelerated horizontal motion superimpose to give the parabolic trajectory of a projectile. If you were lying on the ground you would see the horizontal motion only. If you were watching from a standing position behind the projectile, you would see the vertical motion. Each successive image in both motions occurs after the same periods of time.

Combining the independent vertical and horizontal motions to produce the more complex parabolic trajectory of a projectile.

 

At any point in time, the velocity of the projectile will be the vector sum of the horizontal and vertical components. While the vertical component changes with time, the horizontal component stays the same.

Once the object leaves the table, it experiences a downward acceleration equal to gravity. Thus the vertical velocity (Vy) is continually increasing from zero. The horizontal velocity (Vx) remains constant and is equal to Vxo. The two vectors Vx and Vy are added together to get the velocity at each point on the path. If an object is pointed at an angle, the motion is essentially the same except that there is now an initial vertical velocity (Vyo). Because of the downward acceleration of gravity, Vy continually decreases until it reaches its highest point, at which it begins to fall downward.

 



A simple animation showing how the horizontal and vertical components of velocity change during the trajectory.


When doing problems involving projectiles you should follow the steps below in this order to avoid maing errors:

  1. Resolve the initial velocity into the horizontal and vertical components using trigonometry.

    ux = vcosθ
    uy = vsinθ

  2. Work with the vertical component to calculate time, the vertical component of velocity at that time and the vertical displacement at that time (such as maximum height). The horizontal component of velocity will stay the same because there is no acceleration in the horizontal dimension.

  3. Use vector addition to combine the horizontal and vertical vectors into a velocity vector with magnitude and direction at any point in time. For a fixed initial speed, the maximum range occurs for a 45° launch angle and maximum height occurs for a 90° launch angle.

  4. Calculate the range from the total time of flight.

 

Important Points

For a fixed initial velocity (magnitude), maximum
range occurs for a 45° launch angle and maximum
height occurs for a 90° launch angle.

The following key points will help in your analysis of projectile motion.

  • The time of flight is dependant on the initial vertical velocity of the projectile.

  • The range of a projectile is determined by the initial horizontal velocity and time of flight.

  • The initial horizontal and vertical components of velocity depend on the speed and the angle of elevation. For a fixed initial speed (magnitude), maximum range occurs at 45° while maximum height occurs at 90°.