Introduction to Mechanics Projectiles

Introduction to Mechanics Projectiles Lana heridan De Anza College Feb 6, 2018

Last time relative motion examples

Overview another relative motion example motion with constant acceleration projectiles projectiles with horizontal launch

James. Walker, Physics, 4 Relative Motion Practice (Did th Edition yesterday) You are v riding on a Jet ki at an angle of 35 1 fw, x 1 2.25 m/s upstream on a river tan tan 43 west of north flowing v with a speed 2.40 m/s fw, y of 2.8 m/s. If your velocity relative to the ground is 9.5 m/s at an angle of 20.0 upstream, what is the 2 2 2 2 = x + y = speed of the ( Jet ki ) + relative ( to ) = the water? θ = = = vfw vfw, vfw, 2.25 m/s 2.40 m/s 3.29 m/s er with respect to the ferry, then v fw would have to be shorter and point (Note: Angles are measured relative to the x axis, which points across the river.) r to that depicted in the figure at ski. he flow of the river, such that the et v = bw jet ski s velocity relative to the ground, and v = wg water s 3-8 to find the vector v bw, and then v + v v = v v bw wg bw bg wg ( ) xˆ ( ) ( 8.9 m/s) Ans: xˆ + ( 3.211 m/s m/s ) yˆ 9.5 m/s cos 20.0 + 9.5 m/s sin 20.0 yˆ v v = 1 8.9 Walker, m/s xˆ Physics, + 3.2 m/s yˆ ch 3, 2.8 problem m/s yˆ 55.

Relative Motion Practice For the same river, suppose the Jet ki is moving at a speed of 12 m/s relative to the water. (a) At what angle must you point the Jet ki if your velocity relative to the ground is to be perpendicular to the shore of the river? (b) If you increase the speed of the Jet ki relative to the water, does the angle in part (a) increase, decrease, or stay the same? Explain. (Note: Angles are measured relative to the x axis, which points across the river.) 1 Walker, Physics, ch 3, problem 56.

Kinematic Equations in 2 Dimensions What if velocity is not constant? If there is acceleration, we can still break down our motional quantities (displacement, velocity, acceleration) into components. We can solve each component independently.

r Kinematic Equations in 2 Dimensions same reason, from Figure 4.5b we see that rf is gen i, vi, or a. Finally, notice that vf and rf are gener y v yf a y t v f at y f a y t 2 1 2 v f = v i + at Figure 4.5 Vector representations and components of (a) the velocity and (b) the position of a particle under constant acceleration in two dimensions. a v yi v xi v i v xf a x t x b v yi t y i

r Kinematic Equations in 2 Dimensions same reason, from Figure 4.5b we see that rf is gen i, vi, or a. Finally, notice that vf and rf are gener y v f = v i + at v yf a y t v f at y f a y t 2 1 2 v f = (v x,i i + v y,i j) + (a x i + a y j)t Figure 4.5 Vector representations and components of (a) the velocity and (b) the position of a particle under constant acceleration in two dimensions. v x i + v y j = (v x,i + a x t)i + (v y,i + a y t)j Equating x-components (i-components): a v yi v xi v i v xf a x t x b v yi t y i v x = v x,i + a x t Equating y-components (j-components): v y = v y,i + a y t

nts Motion of Vectors in 2 directions can be resolved into horizontal and Motion in perpendicular directions can be analyzed separately. ents. A vertical force (gravity) does not affect horizontal motion. The horizontal component of velocity is constant. If we know the vertical component of velocity, we can find the time of flight, the maximum height, etc. of the ball, just as before! 1 Drawing by Hewitt, via Pearson.

Acceleration due to gravity and kinematics nts of Vectors Let s think about the components of the motion separately. can be resolved into horizontal and ents. Vertical (y-direction): v y = v i,y gt d y = v i,y t 1 2 gt2 Horizontal (x-direction): a x = 0, v x = v i,x d x = v i,x t 1 Drawing by Hewitt, via Pearson.

Acceleration due to gravity and kinematics A constant acceleration gives motion in the shape of a parabola.

Projectiles projectile Any object that is thrown. We will use this word specifically to refer to thrown objects that experience a vertical acceleration g.

Projectiles projectile Any object that is thrown. We will use this word specifically to refer to thrown objects that experience a vertical acceleration g. For projectile motion, we assume air resistance is negligible. This gives symmetrical parabolic trajectories. Why do we care?

Projectile Velocity 4 v y i y The y component of velocity is zero at the peak of the path. vi u i v xi v y u v v xi v y 0 v The projectile is launched with initial velocity v i. v y g v xi u v v y The x component of velocity remains constant because there is no acceleration in the x direction. v xi u i v x ection 4.2, we stated that two-dimensional motion with constant accele 1 Figure from erway & Jewett, 9th ed.

Projectile Velocity 4 v y i y The y component of velocity is zero at the peak of the path. vi u i v xi v y u v v xi v y 0 The projectile is launched with initial velocity v i. But the y acceleration is not zero! v v y g v xi u v v y The x component of velocity remains constant because there is no acceleration in the x direction. v xi u i v x ection 4.2, we stated that two-dimensional motion with constant accele 1 Figure from erway & Jewett, 9th ed.

Motion of projectiles uppose we launch a projectile from the origin, so that the displacement r = r. Displacement of the object in time t: r = v i t + 1 2 gt2

Projectile s Trajectory erand y d in a y 5 two tion O y vit rf 1 t 2 2 g (x,y) x Figure 4.8 The position vector The object would move in a straight line, but the force of gravity causes it to fall as it moves to the right. r f of a projectile launched from the origin whose initial velocity 1 Figure from erway & Jewett, 9th ed.

volve an algebraic argument. Motion in 2 Dimensions A method of testing that the vectors add as asserted! y Target Cengage Learning/Charles D. Winters Gun 0 u i Line of sight x T Point of collision 1 2 gt 2 x T tan u i y T x b tiflash photograph 1 Figure fromof erway the projectile target & Jewett, 9th ed. demonstration. If the gun

Example Problem uppose the pellet-gun on the previous slide can fire the pellet with an extremely high velocity. (a) How many meters below the line of sight would the pellet be after 5 seconds? (b) If the horizontal component of the pellet s velocity is 20 m/s, how far downrange is the pellet after those 5 seconds? 0 ee Hewitt Conceptual Physics, page 175.

ummary motion with constant acceleration projectiles projectile launched at an angle Homework Walker Physics: Ch 4, onward from page 100. Conceptual Questions: 1, 5; Problems: 3, 5, 7, 9, 11, 13, 15, 17, 25, 84