Projectile Motion Unit 2.6 2 Dimensional Motion Projectile Motion Unit 2.6

Projectile motion We have learned to “resolved” vectors into components. Now we will use the components to understand and predict the motion of objects thrown into the air.

Projectile motion How can you know the displacement, velocity, and acceleration of a ball at any point in time during its flight?

Projectile motion All of the kinematic equations from Unit1 could be rewritten in terms of vector quantities.

Projectile motion When an object is propelled into the air in a direction other than up or down, the velocity, acceleration, and displacement of the object do not all point in the same direction.

Projectile motion This makes the vector forms of the equations difficult to solve.

Projectile motion By resolving vectors into components, you can apply the simpler one dimensional forms of the equations to determine the resultant vectors.

Projectile motion Example: A long jumper in the air has both a horizontal and vertical component.

Projectile motion

Projectile motion In this section, we will focus on the form of 2 dimensional motion called projectile motion. Projectile motion is free fall with an initial horizontal velocity.

Projectile motion Objects that are thrown or launched into the air and are subject to gravity are called projectiles.

Projectile motion The path of a projectile is a curve called a parabola.

Projectile motion Many people mistakenly believe that projectiles eventually fall straight down, much like a cartoon character does after running off a cliff.

Projectile motion However, if an object has initial horizontal velocity in any given time interval, there will be a horizontal motion throughout the flight of the projectile.

Projectile motion We will assume that the horizontal velocity is constant. If we were accounting for air resistance, then velocity would not be constant since a projectile slows down as it collides with air particles.

Projectile motion Hence, the true path of a projectile traveling through Earth’s atmosphere is not a parabola.

Projectile motion To understand the motion a projectile undergoes, let’s examine the following:

Projectile motion The red ball was dropped at the same instant the yellow ball was launched horizontally. If air resistance is disregarded, both balls hit the ground at the same time.

Projectile motion By examining each ball’s position in relation to the horizontal lines and to one another, we see that the two balls fall at the same rate.

Projectile motion The may seem impossible because one is given an interval velocity and the other begins from rest. But if the motion is analyzed one component at a time, it makes sense.

Projectile motion First, consider the red ball that falls straight down. It has no motion in the horizontal direction. In the vertical direction, it starts from rest and proceeds in free fall.

Projectile Motion Thus, the kinematics equation from Unit 1can be applied to analyze the vertical motion of the falling ball.

Projectile motion Note that the acceleration , a, can be rewritten as –g because the only vertical component of acceleration is free-fall acceleration. Note also that Δy is negative.

Projectile motion Vertical motion of a projectile that falls from rest: Δ y = -1/2g (Δt)2 Horizontal motion of a projectile: Δx = vxΔt

Projectile motion Δy = vi(sinθ)Δt -1/2g(Δt)2 Projectiles launched at an angle: Δx= vi(cosθ)Δt Δy = vi(sinθ)Δt -1/2g(Δt)2

Projectile motion To find the velocity of a projectile at any point during its flight, find the vector sum of the components of the velocity at that point.

Projectile motion Use the Pythagorean theorem to find the magnitude of the velocity, and use the tangent function to find the direction of the velocity.