2D Kinematics Projectiles Relative motion

Transcription

1 2D Kinematics Projectiles Relative motion Lana heridan De Anza College Oct 4, 2017

2 Last time 2 dimensional motion projectile motion height of a projectile

3 Overview range of a projectile trajectory equation for a projectile relative motion uniform circular motion

4 Range of a Projectile range The distance in the horizontal direction that a projectile covers ion are before completely hitting the ground. time t as the com- How can we find the range of a projectile? arabolic path ty and accelera- ) nowhere (b) the at what point are to each other? y vi v y 0 tile l case of projectile the origin at t i 5 rns to the same hori- O u i h R Figure 4.9 A projectile launched

5 Range of a Projectile e s motion are completely ely, with time t as the coms in its parabolic path e velocity and accelerather? (a) nowhere (b) the choices, at what point are parallel to each other? y vi v y 0 O ui h R Projectile special case of projectile ed from the origin at t i 5 nd returns to the same hori- Figure 4.9 A projectile launched seballs, over a flat surface from the origin There footballs, is no and acceleration golf in the -direction! (a at t i 5 0 with an initial velocity = 0) aunched. analyze: the peak point, vi. The maimum height of the projectile is h, = and vthe horizontal t, which has coordinates t range is R. At, the peak of the projectile, and the distance trajectory, the particle has coordi- is the ly in terms We just of v i, need u i, and t. g. But tnates (R/2, time h). of flight!

6 Range of a Projectile ely, with time t as the coms in its parabolic path e velocity and accelerather? (a) nowhere (b) the choices, at what point are parallel to each other? y vi v y 0 Projectile special case of projectile ed from the origin at t i 5 nd returns to the same horiseballs, footballs, and golf aunched. analyze: the peak point, t, which has coordinates projectile, and the distance ly in terms of v i, u i, and g. O ui h R Figure 4.9 A projectile launched over a flat surface = vfrom t the origin at t i 5 0 with an initial ( velocity ) vi. The maimum height 2vi sin of the θ R projectile = v i is cos h, and θ the horizontal range is R. At, the peak gof the trajectory, the particle has coordinates = 2v (R/2, i 2 sin θ cos θ h). R g R = v i 2 sin(2θ) g

7 Range of a Projectile A long jumper leaves the ground at an angle of 20.0 above the horizontal and at a speed of 11.0 m/s. How far does he jump in the horizontal direction?

8 Range of a Projectile A long jumper leaves the ground at an angle of 20.0 above the horizontal and at a speed of 11.0 m/s. How far does he jump in the horizontal direction? R = v 2 i sin(2θ) g = (11.0 m/s)2 sin(2 20.0) 9.8 m/s 2 = 7.94 m

9 Projectile Trajectory uppose we want to know the height of a projectile (relative to its launch point) in terms of its coordinate. uppose it is launched at an angle θ above the horizontal, with initial velocity v i.

10 Projectile Trajectory uppose we want to know the height of a projectile (relative to its launch point) in terms of its coordinate. uppose it is launched at an angle θ above the horizontal, with initial velocity v i. For the -direction: = v i cos θt t = v i cos θ

11 Projectile Trajectory uppose we want to know the height of a projectile (relative to its launch point) in terms of its coordinate. uppose it is launched at an angle θ above the horizontal, with initial velocity v i. For the -direction: y-direction: ubstituting for t gives: = v i cos θt t = y = v i sin θt 1 2 gt2 v i cos θ g y = (tan θ) 2vi 2 cos 2 θ 2

12 eight h does a Projectile Motion Eample: Figure P4.24 #25, page A playground is on the flat roof of a city school, 6.00 m above the street below (Fig. P4.25). The vertical wall of the building is h m high, forming a 1-m-high railing around the playground. A ball has fallen to the street below, and a passerby returns it by launching it at an angle of u above the horizontal at a point d m from the base of the building wall. The ball takes 2.20 s to reach a point vertically above the wall. (a) Find the speed at which the ball was launched. (b) Find the vertical distance by which the ball clears the wall. (c) Find the horizontal distance from the wall to the point on the roof where the ball lands. b u d h P erway & Jewett Figure P4.25

13 on P of the particle relative ative Relative to observer Motion B with the we see One that the veryvectors useful rtechnique P A for physical reasoning is considering n other frames of reference. (4.22) A reference frame is a coordinate system that an observer adopts. e, noting that v BA is con- Different observers may have different perspectives: different frames of reference. Consider a pair of observers, one stationary (A), one moving with constant velocity v BA. Both observe a (4.23) W W Galilean velocity particle P. by observer A and transformation up B is its icle velocity rather than v, o reference frames.) Equation equations. They relate A B P bservers in relative motion. When relative velocities are rpa rpb uter ones (P, A) match the t velocities for the particle, We can verify that by taking A vbat B vba Figure 4.20 A particle located

14 Frames of Reference to the other. Q uick Quiz 39.1 How do we relate coordinates in different frames of reference? observer in the Two frames and y O vt y O v P (event) Figure 39.2 An event occurs at Galilean transformations: a point P. The event is seen by two observers in inertial frames and = 9, + where vt, 9 y moves = y with, z a = velocity z, t = t v relative to. uppose some observed by an o frame means tha The event s locati nates (, y, z, t). W of an observer in with uniform rela Consider two in stant velocity v al We assume the or space at some inst and the observer frames in Figure fied location in

15 ummary projectile motion relative motion curving trajectories Quiz start of class, Friday, Oct 6. Collected Homework! due Friday, Oct 6. (Uncollected) Homework erway & Jewett, Ch 4, Work through eample 4.5 (ki Jumper) on page 90. Understand it. (set last time) Ch 4, onward from page 102. Probs: 7, 11, 15, 21, 29, 37, 39 (projectiles, set last time) (Ch 4, onward from page 104. Problems: 40, 43, 45, 51 - relative and circular motion probs, can wait to do)