Question

The final velocity \(v\) of an object in feet per second (ft/s) after it slides down a frictionless inclined plane of height \(h\) feet is $$v=\sqrt{64 h+v_{0}^{2}}$$ where \(v_{0}\) is the initial velocity (in \(\mathrm{ft} / \mathrm{s}\) ) of the object. (a) What is the final velocity \(v\) of an object that slides down a frictionless inclined plane of height 4 feet? Assume that the initial velocity is \(0 .\) (b) What is the final velocity \(v\) of an object that slides down a frictionless inclined plane of height 16 feet? Assume that the initial velocity is \(0 .\) (c) What is the final velocity \(v\) of an object that slides down a frictionless inclined plane of height 2 feet with an initial velocity of \(4 \mathrm{ft} / \mathrm{s} ?\)

Short answer

For part (a), the final velocity is 16 ft/s. For part (b), it is 32 ft/s. For part (c), it is 12 ft/s.

Step-by-step solution

Identify the values given in part (a)

The question provides the height of the inclined plane, which is 4 feet, and the initial velocity, which is 0 ft/s. Let's denote these values as follows:\[ h = 4 \text{ feet} \] \[ v_0 = 0 \text{ ft/s} \]

Plug in the values into the formula for part (a)

Using the formula \[v = \sqrt{64h + v_{0}^{2}}\], we substitute \( h \) and \( v_{0} \) with the values from Step 1: \[ v = \sqrt{64 \times 4 + 0^{2}} \]

Simplify the expression for part (a)

Calculate inside the square root: \[ v = \sqrt{64 \times 4 + 0} = \sqrt{256} = 16 \text{ ft/s} \]

Identify the values given in part (b)

The question provides the height of the inclined plane, which is 16 feet, and the initial velocity, which is 0 ft/s. Let's denote these values as follows: \[ h = 16 \text{ feet} \] \[ v_0 = 0 \text{ ft/s} \]

Plug in the values into the formula for part (b)

Using the formula \[v = \sqrt{64h + v_{0}^{2}}\], we substitute \( h \) and \( v_{0} \) with the values from Step 4: \[ v = \sqrt{64 \times 16 + 0^{2}} \]

Simplify the expression for part (b)

Calculate inside the square root: \[ v = \sqrt{64 \times 16 + 0} = \sqrt{1024} = 32 \text{ ft/s} \]

Identify the values given in part (c)

The question provides the height of the inclined plane, which is 2 feet, and the initial velocity is given as 4 ft/s. Let's denote these values as follows: \[ h = 2 \text{ feet} \] \[ v_0 = 4 \text{ ft/s} \]

Plug in the values into the formula for part (c)

Using the formula \[v = \sqrt{64h + v_{0}^{2}}\], we substitute \( h \) and \( v_{0} \) with the values from Step 7: \[ v = \sqrt{64 \times 2 + (4)^{2}} \]

Simplify the expression for part (c)

Calculate inside the square root: \[ v = \sqrt{64 \times 2 + 16} = \sqrt{128 + 16} = \sqrt{144} = 12 \text{ ft/s} \]

Key concepts

Physics Equations

Physics equations are mathematical representations of physical phenomena. They help explain how different variables interact. In this exercise, we use the equation for final velocity:
\[ v = \sqrt{64h + v_{0}^{2}} \].
This equation tells us how the height (\(h\)) of an inclined plane and the initial velocity (\(v_{0}\)) affect the final velocity (\(v\)) of an object sliding down.
Understanding this equation allows us to calculate real-world behaviors, such as how fast an object moves after sliding down a slope.
You should get comfortable with rearranging and solving equations, as they form the core of problem-solving in physics.

Kinematics

Kinematics is the study of motion without considering the forces causing it. It focuses on quantities like velocity, acceleration, displacement, and time. In our exercise, we work with:

• Initial velocity (\(v_{0}\))
• Height (\(h\))
• Final velocity (\(v\))

Kinematics helps us understand and predict how objects move. By using equations like \(v = \sqrt{64h + v_{0}^{2}}\), we can determine the final velocity of an object just based on its initial conditions and the distance it travels. This is essential for fields like engineering, aviation, and even video game design.

Problem-Solving Steps

Solving physics problems methodically is key to understanding concepts. Let's break down the problem-solving process used in this exercise:

1. **Identify Given Values:** Start by noting what is provided. For instance,\( h = 4 \text{ feet} \) and \( v_{0} = 0 \text{ ft/s} \)
2. **Substitute Values:** Plug these into the equation;\( v = \sqrt{64h + v_{0}^{2}} \)
3. **Simplify:** Solve the equation step-by-step;\( v = \sqrt{64 \times 4 + 0} = 16 \text{ ft/s} \)

Just repeat these steps for different values. Whether the initial velocity is zero or a positive number, following these steps will simplify complex problems.

Velocity Formula

The velocity formula \( v = \sqrt{64h + v_{0}^{2}} \) illustrates how final velocity (\(v\)) is influenced by both the height (\(h\)) and initial velocity (\(v_{0}\)):

- **Height (\(h\)):** Directly affects the term \(64h\). The higher the incline, the more this term contributes to the final velocity.
- **Initial Velocity (\(v_{0}^{2}\)):** This is squared and added to the height term. If an object starts with an initial velocity, this part becomes crucial.

For example, if an object starts with zero initial velocity on a 4-foot incline, the formula simplifies to:
\( v = \sqrt{64 \times 4} = 16 \text{ ft/s} \).
Understanding how to manipulate these variables helps with predicting and analyzing motion in physics.