91Ó°ÊÓ

A skier starts from rest and slides down a mountainside along a straight descending path \(9.0 \mathrm{~m}\) long in \(3.0 \mathrm{~s}\). In what time after starting will the skier acquire a speed of \(24 \mathrm{~m} / \mathrm{s}\) ? Assume that the acceleration is constant and the entire run is straight, at a fixed incline, and around \(1.0\) -km long. We must find the skier's acceleration from the data concerning the 3.0 s trip. Taking the direction of motion down the inclined path as the \(+x\) -direction, we have \(t=3.0 \mathrm{~s}, v_{i x}=0\), and \(x=9.0 \mathrm{~m} .\) Then \(x=v_{i x} t+\frac{1}{2} a t^{2}\) gives $$ a=\frac{2 x}{t^{2}}=\frac{18 \mathrm{~m}}{(3.0 \mathrm{~s})^{2}}=2.0 \mathrm{~m} / \mathrm{s}^{2} $$ We can now use this value of \(a\) for the longer trip, from the starting point to the place where \(v_{f x}=24 \mathrm{~m} / \mathrm{s}\). For this trip, \(v_{i x}=0\), \(u_{f x}=24 \mathrm{~m} / \mathrm{s}, a=2.0 \mathrm{~m} / \mathrm{s}^{2} .\) Then, from \(v_{f}=v_{i}+a t\) $$ t=\frac{v_{f x}-v_{i x}}{a}=\frac{24 \mathrm{~m} / \mathrm{s}}{2.0 \mathrm{~m} / \mathrm{s}^{2}}=12 \mathrm{~s} $$

Step by step solution

01

Identify Known Variables

First, identify the known variables from the problem. We have: initial speed \( v_{i} = 0 \) m/s, distance \( x = 9.0 \) m, time \( t = 3.0 \) s for the initial part of the trip. We need to calculate the acceleration \( a \).

02

Use the Kinematic Equation for Distance

Use the equation for distance with constant acceleration, \( x = v_{i}t + \frac{1}{2} a t^2 \). Since the initial velocity \( v_{i} = 0 \), it simplifies to \( x = \frac{1}{2} a t^2 \).

03

Solve for Acceleration

Rearrange the equation to solve for \( a \): \( a = \frac{2x}{t^2} \). Substitute \( x = 9.0 \) m and \( t = 3.0 \) s to find \( a = \frac{2 \times 9.0}{(3.0)^2} = 2.0 \) m/s\(^2\).

04

Identify Variables for the Longer Trip

For the longer trip, we use the acceleration just calculated with: \( v_{i} = 0 \), final velocity \( v_{f} = 24 \) m/s, and \( a = 2.0 \) m/s\(^2\). We need to find the time \( t \).

05

Use the Kinematic Equation for Final Velocity

Use the equation for final velocity with constant acceleration, \( v_{f} = v_{i} + at \). Since \( v_{i} = 0 \), it simplifies to \( v_{f} = at \).

06

Solve for Time

Rearrange the equation to solve for \( t \): \( t = \frac{v_{f}}{a} \). Substitute \( v_{f} = 24 \) m/s and \( a = 2.0 \) m/s\(^2\) to find \( t = \frac{24}{2.0} = 12 \) s.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinematic Equations

Kinematic equations are invaluable tools in physics, especially when dealing with motion involving constant acceleration. They provide a mathematical framework for predicting various aspects of motion. These equations help relate variables such as time, velocity, acceleration, and displacement. In cases like our skier's problem, they simplify understanding motion over an inclined path.

Here are the primary kinematic equations used in physics:

• Displacement: \( x = v_i t + \frac{1}{2} a t^2 \)
• Final velocity: \( v_f = v_i + a t \)
• Velocity squared: \( v_f^2 = v_i^2 + 2ax \)

Each of these equations is derived from one another under conditions of constant acceleration. In our exercise, we used the first two equations to solve for acceleration and time. Remember, kinematic problems typically necessitate you to identify known variables and choose the appropriate equation to plug these values into. Always verify whether the conditions (like constant acceleration) are satisfied.

Constant Acceleration

Constant acceleration means that the rate of change of velocity remains the same over time. Simply put, an object's velocity changes by the same amount each second. It's a fundamental assumption in the realm of kinematic equations.

For example, in the skier problem, we assumed constant acceleration both when calculating how fast the skier reaches a certain speed and when determining how far they travel within a specific period. Constant acceleration simplifies our calculations as it means the math only involves direct substitution once the right kinematic equation is applied.

When you see 'constant acceleration' in a problem, it tells you:

• The velocity vs. time graph is a straight line.
• Acceleration does not vary with time, making calculations predictable.
• Kinematic equations are reliable as they assume this steadiness.

Understanding this concept is crucial. It not only allows you to handle straightforward tasks but also builds a base for more complex scenarios.

Skier Motion

Analyzing skier motion can be thought of as a typical application problem in physics, often involving basic principles like kinematics and dynamics. In this scenario, the skier moves down an inclined slope, and the problem is simplified by assuming constant acceleration. Practical problems like these help contextualize the theoretical concepts.

When you understand skier motion through physics:

• You see gravity as the key driving force, often simplifying to constant acceleration down the slope.
• You can identify different variables like initial speed, traveled distance, and time.
• You recognize that friction or air resistance is sometimes ignored or presented as negligible, depending on the complexity desired in the problem.

The simplification to a linear path and constant slope allows the use of kinematic equations effectively. Applying these principles, you can predict speeds at various points, understanding crucial factors at play in skier motion dynamics.