1.5 Projectile Motion

A projectile is a "shot" body moving through two or three dimensions. For example, a cannon ball shot into the air is a projectile. When describing the motion of projectiles, it often convenient to break the motion down into one-dimensional components. That is, to speak of the horizontal and vertical (and third-dimensional) motion individually. This is possible because the motion in each direction is independent from that in the others (e.g. vertical motion does not influence horizontal motion).

Gravity is important in projectile motion problems. As will be discussed in Unit 2, a constant unbalanced force, such as a planet’s gravitational force on a free body, results in the body’s constant acceleration. This gravitational acceleration has been proven independent of mass. It has also been found experimentally that the acceleration of gravity near the Earth’s surface is approximately 9.8 m/s². This is useful in the following example:

A soccer ball is placed on the ground and then kicked with an initial velocity of 19.0 m/s at a 30° angle. Calculate how far away from its starting point the ball hits the ground.

We can do this by breaking the ball’s motion into horizontal and vertical components. First, we’ll look at the initial velocity. To calculate the velocity in the horizontal direction, the following steps are taken:

Similarly, to find the vertical component of the velocity, the initial velocity can be multiplied by the sine of 30° to get 9.50 m/s. Now we can find the ball’s traveling time by calculating the times (since we are working with a quadratic position equation, there will be two answers) at which the vertical position of the ball is 0 (it is on the ground).

Remember that velocity is a vector quantity, which means it has a direction. Thus, since the 4.0 m/s is away from the ground and gravitational acceleration is toward the ground, they must have opposite signs. The decision to call the gravitational acceleration negative was arbitrary. However, once sign assignments begin, it is necessary that they be consistent.

After 1.94 s in the air, the ball touches the ground. To find the total horizontal distance traveled in this time, the initial horizontal velocity, which remains constant throughout the flight (since there are no horizontal forces, ignoring air friction), is multiplied by the traveling time:

It is possible to discuss motion in three, four, or any number of dimensions in this manner. We won’t tackle the third dimension right now, but watch for it in the future!

The final equation for this unit follows:

The variable y represents vertical position, x represents horizontal position, v_x0 represents initial horizontal velocity, and g stands for the acceleration of gravity. This equation speeds up the process for solving many two-dimensional projectile motion problems. It is derived by combining vertical and horizontal position equations.