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November 4th, 2019, 08:08 PM  #1 
Senior Member Joined: Jun 2017 From: Lima, Peru Posts: 188 Thanks: 5 Math Focus: Calculus  How can I find the speed of a falling object as seen from a car while in motion?
The problem is as follows: A car driving in a city has a constant velocity of $15\hat{i}15\hat{j}\frac{m}{s}$ with respect to the ground in the highway. However at $t=0\,s$ the driver in the car sees a lady in highest floor of a nearby building dropping a bucket which was at rest. Find the instant (in seconds) from which the driver situated in the car will measure the speed of the bucket to be $15\,i\frac{m}{s}$. You may consider $\vec{g}=10\,\hat{j}\frac{m}{s^{2}}$The alternatives given on my book are as follows: $\begin{array}{ll} 1.&0.5\,s\\ 2.&1\,s\\ 3.&1.5\,s\\ 4.&2\,s\\ 5.&2.5\,s\\ \end{array}$ How am I supposed to relate the quantity of the velocity and the instant?. The only equation which I can recall for the position of an object falling is: $y(t)=y_{0}+v_{0y}t\frac{1}{2}at^2$ But in this case I don't know how tall is the building neither of what else could I do or should I need to find what it is being asked. I suspect that I should sum the vectors for the velocity of the car and that of the bucket falling and from then I could find the time. But I don't know which steps should I take to find such time. Can somebody help me here?. 
November 4th, 2019, 08:31 PM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 3,101 Thanks: 1677 
$v_b = 10 t \, \hat{j} \, m/s$ $v_c = (15 \hat{i}  15 \hat{j}) \, m/s$ $10t \, \hat{j}  (15 \hat{i}  15 \hat{j}) = 15\hat{i} \, m/s$ $15\hat{i}  (10t  15)\hat{j} = 15\hat{i} \, m/s$ $10t 15 = 0 \implies t = 1.5 \, sec$ 

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car, falling, find, motion, object, speed 
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