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Chapter 3: Problem 36
An arrow fired horizontally at \(41 \mathrm{m} / \mathrm{s}\) travels \(23 \mathrm{m}\) horizontally. From what height was it fired?
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The table below lists position versus time for an object moving in the \(x-y\) plane, which is horizontal in this case. Make a plot of position \(y\) versus \(x\) to determine the nature of the object's path. Then determine the magnitudes of the object's velocity and acceleration. $$\begin{array}{ccc} \text { Time, } t(\mathrm{s}) & x(\mathrm{m}) & y(\mathrm{m}) \\ 0 & 0 & 0 \\ 0.10 & 0.65 & 0.09 \\ 0.20 & 1.25 & 0.33 \\ 0.30 & 1.77 & 0.73 \\ 0.40 & 2.17 & 1.25 \\ 0.50 & 2.41 & 1.85 \\ 0.60 & 2.50 & 2.50 \end{array}$$ $$\begin{array}{ccc} \text { Time, } t(\mathrm{s}) & x(\mathrm{m}) & y(\mathrm{m}) \\ 0.70 & 2.41 & 3.15 \\ 0.80 & 2.17 & 3.75 \\ 0.90 & 1.77 & 4.27 \\ 1.00 & 1.25 & 4.67 \\ 1.10 & 0.65 & 4.91 \\ 1.20 & 0.00 & 5.00 \end{array}$$
Find the magnitude of the vector \(34 \hat{\imath}+13 \hat{\jmath} \mathrm{m}\) and determine its angle to the \(x\) -axis.
A particle leaves the origin with its initial velocity given by \(\vec{v}_{0}=11 \hat{\imath}+14 \hat{\jmath} \mathrm{m} / \mathrm{s},\) undergoing constant acceleration \(\vec{a}=-1.2 \hat{\imath}+0.26 \hat{\jmath} \mathrm{m} / \mathrm{s}^{2} .\) (a) When does the particle cross the y-axis? (b) What's its \(y\) -coordinate at the time? (c) How fast is it moving, and in what direction?
Let \(\vec{A}=15 \hat{\imath}-40 \hat{\jmath}\) and \(\vec{B}=31 \hat{\jmath}+18 \hat{k} .\) Find \(\vec{C}\) such that \(\vec{A}+\vec{B}+\vec{C}=\overrightarrow{0}\)
In dealing with nonuniform circular motion, as shown in Fig. 3.23, we should write Equation 3.16 as \(a_{r}=v^{2} / r,\) to show that this is only the radial component of the acceleration. Recognizing that \(v\) is the object's speed, which changes only in the presence of tangential acceleration, differentiate this equation with respect to time to find a relation between the magnitude of the tangential acceleration and the rate of change of the magnitude of the radial acceleration. Assume the radius stays constant.
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