To study 2-D projectile motion, we will analyze two situations: 1. Projectiles launched horizontally 2. Projectiles launched at an angle We will assume there is no air resistance for all examples.
E1. Projectiles launched horizontally Consider a cannonball fired horizontally off of a cliff. We will analyze this motion from the horizontal (x) and the vertical (y) perspectives separately.
Horizontal (x) perspective: Because there is no air resistance, there is no horizontal acceleration. It will move with a constant horizontal velocity (much like a cannonball that experiences no gravity at all).
Vertical (y) perspective: It has no vertical velocity at the start, but it is accelerating downward at 9.81 m/s 2. So, it will speed up vertically (from rest).
Summary (Diagonal Projectiles) Horizontally: Moves at a constant velocity. Use v x = d / t Vertically: Accelerates downward at 9.81 m/s 2 Recall (symmetry) At the same height, v up = v down If it returns to the same height, t up = t down
Ex. 2A golf ball is hit with a speed of 78.0 m/s at 13.0 above the horizontal. Find: a) the maximum horizontal distance the ball travels while it is in the air b) the maximum height of the ball Assume no air resistance and level ground.
Step 1: Find the x- and y-components of the initial velocity y 78.0 m/s v y 13.0 v x x = 76.00 m/s= 17.546 m/s
a) Vertical (y) perspective: From the vertical perspective, the ball starts with an upward velocity of 17.546 m/s. 17.546 m/s It goes up and then comes back down to the same height. 17.546 m/s It will have the same speed when it returns to the same height.
a) Vertical (y) perspective:Ref: Up + ListDown - v i = +17.546 m/s v f = -17.546 m/s a = -9.81 m/s 2 d = t = ? 17.546 m/s We need to find the total time the ball is in the air. 17.546 m/s
a) Vertical (y) perspective:Ref: Up + ListDown - v i = +17.546 m/s v f = -17.546 m/s a = -9.81 m/s 2 d = t = ? Equation: a = v f - v i t
a = v f - v i t a t = v f - v i t = v f - v i a = (-17.546 m/s) - (+17.546 m/s) -9.81 m/s 2 = 3.5772 s
Horizontal (x) Perspective: The ball travels at a constant speed of 76.00 m/s (v x ) for a total time of 3.5772 s. v x = d t d = v x t = (76.00 m/s) (3.5772 s) = 272 m
b) Vertical (y) perspective: When the ball reaches its maximum height, its vertical velocity is zero (even though it is still moving horizontally). Rest 17.546 m/s
b) Vertical (y) perspective:Ref: Up + ListDown - v i = +17.546 m/s v f = 0 Rest a = -9.81 m/s 2 d = t = ? 17.546 m/s Equation: v f 2 = v i 2 + 2 a d
v f 2 = v i 2 + 2 a d v f 2 - v i 2 = 2 a d d = v f 2 - v i 2 2a = 0 - (17.546 m/s) 2 2 (-9.81 m/s 2 ) = 15.7 m