1. Motion in two directions
Projectile Motion
Motion in two directions

2. Key ideas
Horizontal and vertical motion are independent of each other.
Vertical motion is just like free fall, acceleration = g.
Horizontal motion is just like constant velocity, no acceleration.
Algebra

3. Horizontal Component
Ignoring air resistance, there is no net force in the x direction and therefore, no acceleration in the x direction.
The horizontal (x) component of projectile motion is modeled by constant velocity equations, graphs, and motion maps.

4. Horizontal Motion Equations (Constant Velocity)
𝑣 𝑥 = ∆𝑥 ∆𝑡 𝑥= 𝑥 0 + 𝑣 𝑥 𝑡 NO ACCELERATION MEANS 𝑣 0𝑥 = 𝑣 𝑥

5. Vertical Component
There is a net force called weight or gravity acting in the y direction, so there must be an acceleration due to gravity in the y direction.
The vertical (y) component of projectile motion is modeled by constant acceleration equations, graphs, and motion maps.

6. Vertical Motion Equations (Constant Acceleration)
𝑎 𝑦 = ∆ 𝑣 𝑦 ∆𝑡 𝑣 𝑦 = 𝑣 0𝑦 + 𝑎 𝑦 𝑡 ∆𝑦= 1 2 𝑎 𝑦 𝑡 2 + 𝑣 𝑜𝑦 𝑡 𝑣 𝑦 2 = 𝑣 0𝑦 2 + 2𝑎 𝑦 ∆𝑦 𝑎 𝑦 =𝑔=9.8 𝑚 𝑠 2 𝑜𝑛 𝐸𝑎𝑟𝑡ℎ

7. Concept Check
Two spheres of equal mass, A and B, are projected off the edge of a bench 2.0 m high. Sphere A has a horizontal velocity of 5.0 m/s and sphere B has a horizontal velocity of 2.5 m/s. If they both leave the bench at the same instant, which will land first? Which will land a greater distance from the edge of the table?

8. Concept Check
Two spheres, A and B, are projected off the edge of a bench 2.0 m high with the same horizontal velocity. Sphere A has a mass of 10 g and sphere B has a mass of 5 g. If they both leave the bench at the same instant, which will land first? Which will land a greater distance from the edge of the table?

9. Concept Check
Two spheres A and B, where A has twice the mass of B, are projected off the edge of two different height shelves. Sphere A leaves from a height of 2 m and sphere B leaves from a height of 1 m. If they both leave the bench at the same instant, which will land first? Which will land a greater distance from the edge of the table?

10. Equations 𝑣= ∆𝑥 ∆𝑡 𝑥=𝑣𝑡+ 𝑥 0
Horizontal (x)
Vertical (y)
𝑣= ∆𝑥 ∆𝑡 𝑥=𝑣𝑡+ 𝑥 0
𝑎= ∆𝑣 ∆𝑡 𝑣=𝑎𝑡+ 𝑣 0 ∆𝑦= 1 2 𝑔 𝑡 2 + 𝑣 0 𝑡 𝑣 𝑦 2 = 𝑣 0𝑦 2 + 2𝑎 𝑦 ∆𝑦

12. Example
A ball rolls off a 1.0 m high table and lands on the floor 3.0 m away from the table.
How long is the ball in the air?

13. Example
A ball rolls off a 1.0 m high table and lands on the floor 3.0 m away from the table.
B. With what horizontal velocity did the ball leave the table?

14. Example
A ball rolls off a 1.0 m high table and lands on the floor 3.0 m away from the table.
C. What is the vertical velocity of the ball just before it hits the floor?

15. Example
A ball rolls off a 1.0 m high table and lands on the floor 3.0 m away from the table.
D. What is the horizontal velocity of the ball just before it hits the floor?