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Chapter 3: Problem 82

A diver jumps from a \(40.0 \mathrm{~m}\) high cliff into the sea. Rocks stick out of the water for a horizontal distance of \(7.00 \mathrm{~m}\) from the foot of the cliff. With what minimum horizontal speed must the diver jump off the cliff in order to clear the rocks and land safely in the sea?

Short Answer

Expert verified
Answer: The minimum horizontal speed required for the diver to clear the rocks and land safely is approximately 2.44 m/s.

Step by step solution

01

Calculate time taken in vertical motion

To calculate the time taken for the diver to fall 40 meters, we can use the following kinematic equation: \(h = ut + \frac{1}{2}gt^2\) where \(h\) is the vertical distance fallen (40 meters), \(u\) is the initial vertical velocity (0, since the diver jumps off horizontally), \(g\) is the acceleration due to gravity (\(9.81 \mathrm{m/s^2}\)), and \(t\) is the time taken. Plugging in the values, we get: \(40 = 0 \cdot t + \frac{1}{2}(9.81)t^2\)
02

Solve for time

We can solve this equation for \(t\), the time taken for the diver to fall 40 meters: \(40 = \frac{1}{2}(9.81)t^2\) To find the value of \(t\), we can first multiply both sides of the equation by 2: \(80 = (9.81)t^2\) Now dividing both sides by 9.81: \(t^2 = \frac{80}{9.81}\) Taking the square root: \(t = \sqrt{\frac{80}{9.81}}\) Using a calculator, we find that: \(t \approx 2.87 \mathrm{s}\)
03

Calculate required horizontal speed

Now that we have the time taken for the diver to fall 40 meters, we can find the required horizontal speed to clear the 7 meters of rocks. We'll use the following equation for horizontal motion: \(d = vt\) where \(d\) is the horizontal distance (7 meters), \(v\) is the horizontal speed, and \(t\) is the time calculated in step 2 (\(2.87 \mathrm{s}\)). Plugging in the values: \(7 = v(2.87)\) Now, we can solve for \(v\): \(v = \frac{7}{2.87}\) Using a calculator, we find the required horizontal speed: \(v \approx 2.44 \mathrm{m/s}\) The diver must jump off the cliff with a minimum horizontal speed of \(2.44 \mathrm{m/s}\) to clear the rocks and land safely in the sea.

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Most popular questions from this chapter

A particle's motion is described by the following two parametric equations: $$ \begin{array}{l} x(t)=5 \cos (2 \pi t) \\ y(t)=5 \sin (2 \pi t) \end{array} $$ where the displacements are in meters and \(t\) is the time, in seconds. a) Draw a graph of the particle's trajectory (that is, a graph of \(y\) versus \(x\) ). b) Determine the equations that describe the \(x\) - and \(y\) -components of the velocity, \(v_{x}\) and \(v_{y}\), as functions of time. c) Draw a graph of the particle's speed as a function of time.

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