Two Dimensional Motion

Student Expectation

The student is expected to analyze and describe accelerated motion in two dimensions using equations, including projectile and circular examples.

Key Concepts

• • Both projectile motion and circular motion are examples of accelerated motion in two dimensions. Projectile motion can be described as motion of an object when gravity is the only force acting on the object. Circular motion can be described as an object traveling around the circumference of a circle or in a circular orbit.

  • Projectile motion can be analyzed using two dimensional vector equations, one for acceleration due to gravity and the other for horizontal speed.

  • Circular motion can be analyzed in two dimensions from an equation involving the radius of the object from a certain point and the time for the object to revolve around the point once.

  • Acceleration in two dimensions means the object’s velocity is changing, either in direction or in magnitude or both.

TWO DIMENSIONAL MOTION

Projectile Motion

Two dimensional motion can be described using the two separate components. The two separate motions are in horizontal and vertical directions respectively. Projectile motion is two-dimensional because it has a horizontal component and a vertical component. These components can be examined separately as one-dimensional motion, then put together to get the whole picture of what is happening. After the projectile has been launched, the only force acting on it is gravity. The gravitational force pulls the projectile toward the ground and accelerates it as it falls. This means that the projectile is a special case of free-fall and can be treated as a free-fall problem in the vertical (y) direction. In the horizontal direction, there are no forces acting on the projectile. It moves at a constant speed as it travels through the air. The initial speed in the horizontal (x) direction determines how far the projectile travels. This is known as its range.

Projectile Motion Equations

The kinematic equations of motion that apply to one-dimensional motion can be used for projectiles because they can be applied to the horizontal direction and the vertical direction separately. There is no acceleration in the horizontal direction so a complex displacement formula becomes a simple constant velocity equation that can be solved for horizontal displacement, time, or velocity.

In the vertical component, there is no initial velocity, so the displacement formula becomes a simple constant acceleration equation as shown below:

If we know the equations that express the motion in the horizontal and the vertical directions, these equations can be utilized to calculate accelerated motion in two dimensions for projectiles.

In projectile motion, the velocity can be calculated by using vertical velocity and horizontal velocity as listed above. It can be analyzed using two dimensional vector equations, one for acceleration due to gravity and the other for horizontal speed, which is determined by the initial velocity in the horizontal direction.

Circular Motion

Circular motion is also a form of two-dimensional motion, which occurs when an object travels around a circular path. A special case of circular motion is uniform circular motion. An object moving with uniform circular motion travels around a circular path at a constant speed. A Ferris wheel, merry-go-round, and the rotating hands on an analog watch are examples of circular motion. Some objects exhibit uniform circular motion even if they do not travel in a full circle. For example, when a car enters a curved exit ramp, it may travel around a portion of a circular path, but it still undergoes circular motion during that time.

Circular Motion Equations

We can calculate the average speed, v, of an object traveling in uniform circular motion around a circle of radius, r. To do this, recall that the average speed of an object in one-dimensional motion is equal to the total distance traveled divided by the total time of the journey:

When an object travels around a circular path, the total distance that it travels in one complete loop is equal to the circumference of the circle. The equation for the circumference, C, of a circle of radius, r, is as follows:

C = 2πr = distance

We can define the time it takes for an object to complete one full loop around the circle as T, which is also called the period of the motion. We can plug these variables into the equation for average speed to determine the equation for average speed in uniform circular motion:

Although the object moves with constant speed, its direction is constantly changing. Thus, the object is accelerating. The acceleration of an object in uniform circular motion is called centripetal acceleration. This vector points inward, toward the center of the circle. The magnitude of centripetal acceleration, a_c, can be calculated using the following equation:

In uniform circular motion, the velocity, v (called tangential velocity), is always tangent to the path of motion. Centripetal acceleration (a_c) is always perpendicular to the velocity and points toward the center of the circle as shown below: