91Ó°ÊÓ

A skier starts from rest and slides down a mountain side 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 and at a fixed incline. We must find the skier's acceleration from the data concerning the \(3.0 \mathrm{~s}\) 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, \(v_{i x}=0, v_{f x}=24 \mathrm{~m} / \mathrm{s}, a=2.0 \mathrm{~m} / \mathrm{s}^{2} .\) 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 values for initial trip

We have a skier starting from rest, so the initial velocity \( v_{i x} = 0 \) m/s. The skier covers a distance of \( x = 9.0 \) m in a time of \( t = 3.0 \) s.

02

Use the kinematic equation to find acceleration

The kinematic equation \( x = v_{i x} t + \frac{1}{2} a t^{2} \) can be rearranged to solve for acceleration \( a \). Since \( v_{i x} = 0 \), the equation simplifies to \( x = \frac{1}{2} a t^{2} \). Solving for \( a \) gives:\[ a = \frac{2x}{t^{2}} = \frac{2 \times 9.0 \, \mathrm{m}}{(3.0 \, \mathrm{s})^{2}} = \frac{18 \, \mathrm{m}}{9 \, \mathrm{s}^{2}} = 2.0 \, \mathrm{m/s^{2}} \]

03

Identify known values for longer trip to target speed

For the longer trip, the skier starts from rest again \( v_{i x} = 0 \) and needs to reach a final velocity of \( v_{f x} = 24 \, \mathrm{m/s} \) using the previously calculated acceleration \( a = 2.0 \, \mathrm{m/s^2} \).

04

Use velocity equation to find time to reach target speed

Using the equation for velocity, \( v_{f} = v_{i} + at \), we can solve for \( t \) when \( v_{f x} = 24 \, \mathrm{m/s} \):\[ t = \frac{v_{f x} - v_{i x}}{a} = \frac{24 \, \mathrm{m/s} - 0 \, \mathrm{m/s}}{2.0 \, \mathrm{m/s^2}} = 12 \, \mathrm{s} \]

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Key Concepts

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

Acceleration

Acceleration is a key concept in kinematics. It can be thought of as how quickly something speeds up or slows down. Mathematically, acceleration is defined as the rate of change of velocity over time. If a skier starts from rest, this means they begin from a state where their velocity is zero, yet accelerate as they move downhill. In this exercise, we calculated acceleration using the kinematic equation:

• First, knowing the skier travels 9 meters in 3 seconds, we acknowledged the initial velocity is zero.
• With the equation for motion: \( x = v_{i} \, t + \frac{1}{2} a \, t^{2} \), and since \( v_i = 0 \), it simplifies to \( x = \frac{1}{2} a \, t^{2} \).
• Rearranging, we solved for acceleration: \( a = \frac{2x}{t^2} \).

This determination of acceleration is crucial, as it allows us to understand how fast the skier will be moving at any moment.

Velocity

Velocity is another fundamental concept in kinematics and indicates the speed of an object in a specific direction. It differs from speed as it considers direction as well. In this problem, we deal with initial and final velocities.

• Initial velocity \( v_{i} \) is the speed at which the skier starts. Since they begin from rest, \( v_{i} = 0 \; \text{m/s} \).
• Final velocity \( v_{f} \) represents the speed of the skier after a certain time or distance. For the skier to reach a target speed of 24 m/s, we use known acceleration values to calculate the required time.

By understanding how velocity changes with time and under the influence of acceleration, we can predict motion beyond initial assumptions.

Kinematic Equations

Kinematic equations are instrumental in solving problems related to motion in physics. They provide relationships between displacement, initial velocity, final velocity, acceleration, and time. Critical equations used in this exercise include:

• \( x = v_{i}t + \frac{1}{2}at^{2} \): Used to calculate how far an object moves when starting with an initial velocity.
• \( v_{f} = v_{i} + at \): Used to find out an object's final velocity after accelerating over time.

These equations assume constant acceleration and a straight path of motion, simplifying the calculations required to determine unknown variables, such as time or velocity.

Constant Acceleration

Assuming constant acceleration is often a simplifying assumption in physics, making it easier to apply kinematic equations as derived from the laws of motion. Constant acceleration means that the rate of change of velocity is steady over time. In this scenario, the skier accelerating downhill maintains a constant rate of acceleration.

• This allows us to use linear equations since changes are predictable and uniform.
• In the context of the problem, knowing acceleration is constant enables straightforward calculations of time to reach certain velocities.

By understanding the assumption of constant acceleration, we streamline problem-solving, leading to more accurate and efficient calculations in motion-related questions.