Answer:
32.66 m/s
Explanation:
![k=30000N/m\\x=4 cm=0.04 m\\m=45g=0.045kg](https://tex.z-dn.net/?f=k%3D30000N%2Fm%5C%5Cx%3D4%20cm%3D0.04%20m%5C%5Cm%3D45g%3D0.045kg)
The potential energy stored in the spring will impart kinetic energy into the dart:
![\frac{1}{2}kx^2=\frac{1}{2}mv^2\\v=\sqrt{\frac{k}{m}}x](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7Dkx%5E2%3D%5Cfrac%7B1%7D%7B2%7Dmv%5E2%5C%5Cv%3D%5Csqrt%7B%5Cfrac%7Bk%7D%7Bm%7D%7Dx)
substitute the values:
![v=\sqrt{\frac{30000}{0.045}}\times0.0=32.66m/s](https://tex.z-dn.net/?f=v%3D%5Csqrt%7B%5Cfrac%7B30000%7D%7B0.045%7D%7D%5Ctimes0.0%3D32.66m%2Fs)
Answer:
accelerated motion
Explanation:
a change in velocity (10 m/s to 50 m/s) over time (5 s) is called acceleration.
40/5 = 8 m/s²
Answer: It should be the 3rd option down!
Explanation:
The whole point of this problem is to check how well you understand
the definitions of a few important quantities, like velocity, speed, distance,
displacement etc.
Before we begin, I just want to mention that 'MPG' is not a unit of either
velocity or speed, but I think I know what you mean.
-- For some reason, Ms. Eaddy rode 100 miles north on the train, then
stayed aboard while the train turned around and took her 150 miles south.
The total distance she rode was (100 + 150) = 250 miles. But she ended up
50 miles south of where she began.
-- Displacement for the whole trip = distance and direction from the start point
to the finish point.
Displacement = 50 miles south
-- Average velocity = (displacement) / (time)
50 miles south / 3.5 hours = <u>14.29 miles per hour south</u>
Explanation:
(a) After the engines stop, the rocket reaches a maximum height at which it will stop and begin to descend in free fall due to gravity.
(b) We must separate the motion into two parts, when the rocket's engines is on and when the rocket's engines is off.
First we must find the rocket speed when the engines stop:
![v_f^2=v_0^2+2ay_1\\v_f^2=(52\frac{m}{s})^2+2(1\frac{m}{s^2})(160m)\\v_f^2=3024\frac{m^2}{s^2}\\v_f=\sqrt{3024\frac{m^2}{s^2}}=54.99\frac{m}{s}](https://tex.z-dn.net/?f=v_f%5E2%3Dv_0%5E2%2B2ay_1%5C%5Cv_f%5E2%3D%2852%5Cfrac%7Bm%7D%7Bs%7D%29%5E2%2B2%281%5Cfrac%7Bm%7D%7Bs%5E2%7D%29%28160m%29%5C%5Cv_f%5E2%3D3024%5Cfrac%7Bm%5E2%7D%7Bs%5E2%7D%5C%5Cv_f%3D%5Csqrt%7B3024%5Cfrac%7Bm%5E2%7D%7Bs%5E2%7D%7D%3D54.99%5Cfrac%7Bm%7D%7Bs%7D)
This final speed is the initial speed in the second part of the motion, when engines stop until reach its maximun height. Therefore, in this part the final speed its zero and the value of g its negative, since decelerates the rocket:
![v_f^2=v_0^2+2gy_{2}\\y_{2}=\frac{v_f^2-v_0^2}{2g}\\y_{2}=\frac{0^2-(54.99\frac{m}{s})^2}{2(-9.8\frac{m}{s^2})}=154.28m](https://tex.z-dn.net/?f=v_f%5E2%3Dv_0%5E2%2B2gy_%7B2%7D%5C%5Cy_%7B2%7D%3D%5Cfrac%7Bv_f%5E2-v_0%5E2%7D%7B2g%7D%5C%5Cy_%7B2%7D%3D%5Cfrac%7B0%5E2-%2854.99%5Cfrac%7Bm%7D%7Bs%7D%29%5E2%7D%7B2%28-9.8%5Cfrac%7Bm%7D%7Bs%5E2%7D%29%7D%3D154.28m)
So, the maximum height reached by the rocket is:
![h=y_1+y_2\\h=160m+154.28m=314.28m](https://tex.z-dn.net/?f=h%3Dy_1%2By_2%5C%5Ch%3D160m%2B154.28m%3D314.28m)
(c) In the first part we have:
![v_f=v_0+at_1\\t_1=\frac{v_f-v_0}{a}\\t_1=\frac{54.99\frac{m}{s}-52\frac{m}{s}}{1\frac{m}{s^2}}\\t_1=2.99s](https://tex.z-dn.net/?f=v_f%3Dv_0%2Bat_1%5C%5Ct_1%3D%5Cfrac%7Bv_f-v_0%7D%7Ba%7D%5C%5Ct_1%3D%5Cfrac%7B54.99%5Cfrac%7Bm%7D%7Bs%7D-52%5Cfrac%7Bm%7D%7Bs%7D%7D%7B1%5Cfrac%7Bm%7D%7Bs%5E2%7D%7D%5C%5Ct_1%3D2.99s)
And in the second part:
![t_2=\frac{v_f-v_0}{g}\\t_2=\frac{0-54.99\frac{m}{s}}{-9.8\frac{m}{s^2}}\\t_2=5.61s](https://tex.z-dn.net/?f=t_2%3D%5Cfrac%7Bv_f-v_0%7D%7Bg%7D%5C%5Ct_2%3D%5Cfrac%7B0-54.99%5Cfrac%7Bm%7D%7Bs%7D%7D%7B-9.8%5Cfrac%7Bm%7D%7Bs%5E2%7D%7D%5C%5Ct_2%3D5.61s)
So, the time it takes to reach the maximum height is:
![t_3=t_1+t_2\\t_3=2.99s+5.61s=8.60s](https://tex.z-dn.net/?f=t_3%3Dt_1%2Bt_2%5C%5Ct_3%3D2.99s%2B5.61s%3D8.60s)
(d) We already know the time between the liftoff and the maximum height, we must find the rocket's time between the maximum height and the ground, therefore, is a free fall motion:
![v_f^2=v_0^2+2ay\\v_f^2=0^2+2(9.8\frac{m}{s^2})(314.28m)\\v_f=\sqrt{6159.888\frac{m^2}{s^2}}=78.48\frac{m}{s}](https://tex.z-dn.net/?f=v_f%5E2%3Dv_0%5E2%2B2ay%5C%5Cv_f%5E2%3D0%5E2%2B2%289.8%5Cfrac%7Bm%7D%7Bs%5E2%7D%29%28314.28m%29%5C%5Cv_f%3D%5Csqrt%7B6159.888%5Cfrac%7Bm%5E2%7D%7Bs%5E2%7D%7D%3D78.48%5Cfrac%7Bm%7D%7Bs%7D)
![t_4=\frac{v_f-v_0}{g}\\t_4=\frac{78.48\frac{m}{s}-0}{9.8\frac{m}{s^2}}\\t_4=8.01s](https://tex.z-dn.net/?f=t_4%3D%5Cfrac%7Bv_f-v_0%7D%7Bg%7D%5C%5Ct_4%3D%5Cfrac%7B78.48%5Cfrac%7Bm%7D%7Bs%7D-0%7D%7B9.8%5Cfrac%7Bm%7D%7Bs%5E2%7D%7D%5C%5Ct_4%3D8.01s)
So, the total time is:
![t=t_3+t_4\\t=8.60s+8.01s\\t=16.61s](https://tex.z-dn.net/?f=t%3Dt_3%2Bt_4%5C%5Ct%3D8.60s%2B8.01s%5C%5Ct%3D16.61s)