Position, Velocity and Acceleration in 2D

1. Position, Velocity and Acceleration in 2D Position in 2D Displacement in 2D Velocity in 2D Acceleration in 2D There are no changes to the definitions of these quantities, but we must remember that there are two components for each vector!!

2. 2D Motion with Constant Acceleration Differentiating vectors The constant acceleration equations are derived in an identical fashion to the way they were derived in 1D. They will not be re-derived here, but rewritten to include 2D. The subscripts show the direction we are working with, so we can ignore the unit vectors while working with the equations in this form. or and and or The x and y directions are completely independent of each other!!

3. Example: A particle moves in a horizontal xy-plane. It’s x and y positions as a function of time are given below. Write an expression for the a) position, b) velocity and c) acceleration in vector form. a) b) c)

4. Projectile Motion – an application of 2D motion We will use 2D motion to model the trajectory (path) a projectile follows as it travels through the air. • We are going to use two assumptions to simplify the problem: • g is a constant over the range of motion. • This is accurate as long as the range of motion is small compared to the radius of the Earth. • Air resistance is negligible. • This is Not justified for any real world example, but it is close for low speeds. • We are considering an ideal case to learn the basics. The trajectory is modeled using horizontal and vertical position. Both the horizontal and vertical positions change in time. What do these assumptions tell us about the vertical and horizontal motion? Vertical motion – Constant acceleration – the vertical motion is parabolic in time. Horizontal motion – Zero acceleration – the horizontal motion is linear with time and the velocity is constant. Since the accelerations for each direction of motion are different we can look at the two different directions of motion separately. How do we link the two motions? Both vertical and horizontal motion occurs simultaneously, therefore we can link the two distinct motions with time.

5. x y t t V Vy y Vx What would this motion look like if you rotated the x-axis so it was directed towards you? What would this motion look like if you looked down on it from above? y y x The actual motion is a combination of both of these motions occurring simultaneously. We can treat the vertical and horizontal motions as separate 1D motion. x We use the same 1D constant acceleration motion equations to model 2D constant acceleration motion. x

6. It is also sometimes useful to model the trajectory of a projectile by finding an equation that provides the vertical position based on the horizontal position. We will assume a known initial velocity v0 and a known launch angle q. 0 0 0 Equation of a parabola This shows us that the trajectory of an object will be parabolic. Remember the initial assumptions we made are still being used.

7. A funnel cart, as seen in the photograph below, consists of a cart with a funnel mounted on it. It is constructed with frictionless wheels (We can do this in physics.) so that it will roll with a constant speed along the straight level track in the photograph. A spring in the funnel can be compressed so that if a ball is placed in the funnel, and the funnel then compressed and released, the ball will be ejected directly up and fall back into the funnel. The ball is HEAVY, so air resistance is negligible. When the cart is pushed from left to right along the track, a trip on the track hits a cam connected to the funnel, ejecting the ball when the cart gets to the location at which it is shown in the picture below, as the cart moves along the track. When the ball is ejected in this manner, where will it fall? (1) The ball will fall in front of the funnel. (2) The ball will fall behind the funnel. (3) The ball will fall IN the funnel.