Question

12. The height and speed of a projectile (such as a thrown ball) launched with a speed of \(v_{0}\) at an angle \(A\) to the horizontal are given by

$$ \begin{array}{c} h(t)=v_{0} t \sin A-0.5 g t^{2} \\ v(t)=\sqrt{v_{0}^{2}-2 v_{0} g t \sin A+g^{2} t^{2}} \end{array} $$

where \(g\) is the acceleration due to gravity. The projectile will strike the ground when \(h(t)=0\), which gives the time to hit \(t_{\text {hit }}=2\left(v_{0} / g\right) \sin A\). Suppose that \(A=30^{\circ}, v_{0}=40 \mathrm{~m} / \mathrm{s}\), and \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\). Use the

MATLAB relational and logical operators to nd the times when

a. The height is no less than \(15 \mathrm{~m}\).

b. The height is no less than \(15 \mathrm{~m}\) and the speed is simultaneously no greater than \(36 \mathrm{~m} / \mathrm{s}\).

c. The height is less than \(5 \mathrm{~m}\) or the speed is greater than \(35 \mathrm{~m} / \mathrm{s}\).

Answer 1

It is helpful to plot the height and speed versus time before answering the questions specially for part c The plot is shown in the figure below

40 : Speed (m/s) : 25F.. Height (m) 0 2.5 3 3.5 4 4.5 2 Time (sec) 0 0.5 1.5 The rest of the following script file can be used to compute the answers with more precision than can be obtained by reading the plot. VO40;g 9.81 A- 30 pi/180; t_hit-2VO sin(A)g; hvO t*sin(A)-0.5*g t. 2; plot(t,h,t,v),xlabel( Time (sec)), gtext(Height),gtext('Speed'),grid % Part (a) ua- find(h-15);

tla (ua(1)-1) (t_hit/100) t2aua(length(ua)-1)-(t hit/100) % Part (b). ub find(h-15&v< 36); t1b (ub(1-1)-(t hit/100) t2b-ub(length(ub)-1)-(t hit/100) % Part (c). tic (uc(1)-1) (t_hit/100) t2cuc(length(uc)-1) (t hit/100) When run, this file produces the results: t1a-1.0194, 12a 3.0581, t1b 1.0601, 12b-3.0173, tic 1.5494, t2c 2.5280. Thus the height is no less than 15 meters for 1.0194 s ts 3.0581 seconds. The height is no less than 15 meters and the speed is simultaneously no greater than 36 meters/second for 1.0601 sts 3.0173 seconds. The height is less than 5 meters or the speed is greater than 35 meters/second for 1.5494 s ts 2.528 seconds.