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Chapter 5: Problem 16

A skier starts from rest at the top of a \(24^{\circ}\) slope \(1.3 \mathrm{km}\) long. Neglecting friction, how long does it take to reach the bottom?

Short Answer

Expert verified
To find the time taken to reach the bottom of the slope, first calculate the effective acceleration along the slope due to gravity and then use that to find the time using the equation of motion.

Step by step solution

01

Identification of Known Variables

Firstly, from the question, it is known that the initial velocity (\(v_0\)) is zero (as the skier starts from rest), the displacement (\(d\)) is 1.3 km or 1300 meters (the length of the slope) and acceleration due to gravity (\(g\)) is \(9.8 \, m/s^2\). The angle of the slope (\(\theta\)) is given as \(24^{\circ}\).
02

Calculation of Acceleration On the Slope

The acceleration of the skier along the slope (\(a\)) is given by \( a = g \cdot \sin(\theta)\). Use this equation to find the acceleration. Be sure to convert the angle from degrees to radians before using it in this formula. The sin of \(24^{\circ}\) is approximately 0.4067.
03

Calculate the Time

The next step is to calculate the time it takes to reach the bottom. To do this, use the formula \(d = v_0t + 0.5at^2\), where \(d\) is the distance, \(v_0\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time. Since \(v_0\) is zero, the formula becomes \(d = 0.5at^2\). Solve for \(t\) to find the time it takes to reach the bottom.

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Most popular questions from this chapter

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