91Ó°ÊÓ

A large boulder is ejected vertically upward from a volcano with an initial speed of \(40.0 \mathrm{~m} / \mathrm{s}\). Air resistance may be ignored. (a) At what time after being ejected is the boulder moving at \(20.0 \mathrm{~m} / \mathrm{s}\) upward? (b) At what time is it moving at \(20.0 \mathrm{~m} / \mathrm{s}\) downward? (c) When is the displacement of the boulder from its initial position zero? (d) When is the velocity of the boulder zero? (e) What are the magnitude and direction of the acceleration while the boulder is (i) moving upward? (ii) moving downward? (iii) at the highest point? (f) Sketch graphs of \(a_{y}\) versus \(t, v_{y}\) versus \(t,\) and \(y\) versus \(t\)

Step by step solution

01

Identify Given Information and Constants

Initial speed of the boulder, \( v_0 = 40.0 \, \text{m/s} \). We know that air resistance is ignored, so the only force acting vertically on the boulder is gravity. Therefore, the acceleration, \( a = -9.8 \, \text{m/s}^2 \), as it acts downward.

02

Solving for Time Moving Upward at 20.0 m/s (Part A)

Use the formula \( v = v_0 + at \). Here, \( v = 20.0 \, \text{m/s} \), \( v_0 = 40.0 \, \text{m/s} \), and \( a = -9.8 \, \text{m/s}^2 \).\[20.0 = 40.0 - 9.8t\]Solving for \( t \):\[9.8t = 40.0 - 20.0 \t = \frac{20.0}{9.8} \approx 2.04 \, \text{s}\]

03

Solving for Time Moving Downward at 20.0 m/s (Part B)

For downward motion at \( 20.0 \, \text{m/s} \), set \( v = -20.0 \, \text{m/s} \) (since direction is down). Use the equation again:\[-20.0 = 40.0 - 9.8t\]Solving for \( t \):\[9.8t = 40.0 + 20.0 \t = \frac{60.0}{9.8} \approx 6.12 \, \text{s}\]

04

Finding Time When Displacement is Zero (Part C)

The total time for zero displacement is double the time it takes to reach the peak. At the peak, speed \( v = 0 \). Use:\[0 = 40.0 - 9.8t\]Solving for \( t \):\[t = \frac{40.0}{9.8} \approx 4.08 \, \text{s} \]Therefore, displacement is zero at \( 2 \times 4.08 = 8.16 \, \text{s} \).

05

Finding Time When Velocity is Zero (Part D)

The velocity is zero at the highest point. We already found this in Step 4:\[t = 4.08 \, \text{s}\]

06

Acceleration Analysis (Part E)

Since air resistance is negligible, the acceleration at all points is gravity. Thus, \( a = -9.8 \, \text{m/s}^2 \):(i) Moving upward: \(-9.8 \, \text{m/s}^2\)(ii) Moving downward: \(-9.8 \, \text{m/s}^2\)(iii) At the highest point: \(-9.8 \, \text{m/s}^2\)

07

Sketching Graphs (Part F)

For each graph:- **Acceleration vs Time (\(a_y\)):** A horizontal line at \(-9.8 \, \text{m/s}^2\).- **Velocity vs Time (\(v_y\)):** A straight line decreasing from \(40.0 \, \text{m/s}\) to \(0\) at \(4.08 \, \text{s}\), then continuing to \(-40.0 \, \text{m/s}\) at \(8.16 \, \text{s}\).- **Position vs Time (\(y\)):** A parabola opening downward, reaching its peak at \(4.08 \, \text{s}\) and returning to zero at \(8.16 \, \text{s}\).

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Key Concepts

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

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. It focuses on displacement, velocity, and acceleration. In this exercise, we are dealing with a boulder being ejected upwards, which makes it an example of two-dimensional kinematics.

Key terms in kinematics include:

• Displacement: The change in position of an object. For the boulder, its displacement at various times is crucial to solving the exercise.
• Velocity: The speed of the object in a given direction. Initial velocity is given, and changes over time due to acceleration.
• Acceleration: The rate of change of velocity. Here, it is due to gravity, acting downwards.

Understanding kinematics allows prediction of future positions and velocities of the boulder. Since air resistance is neglected, calculations are simplified, relying solely on gravitational forces.

Vertical Motion

In physics, vertical motion refers to the movement of an object along the y-axis, or vertically, under the influence of gravity. When an object is projected vertically like our boulder from a volcano, its speed and position change as it moves upward and then downward.

For vertical motion:

• The velocity of an object reduces as it moves against gravity, reaching zero at the highest point.
• After reaching the peak, the object accelerates downward due to gravity.

The equation of motion used is: \[ v = v_0 + at \]This relates the final velocity \( v \), initial velocity \( v_0 \), acceleration \( a \), and time \( t \). In the exercise, this helps us find when the boulder's velocity reaches specified values. Understanding vertical motion simplifies the analysis of objects like our boulder moving up and then falling back down.

Free Fall

Free fall is defined as the motion of an object under the influence of gravitational force only. No other forces, such as air resistance, are considered. This principle applies to the boulder's movement once it is ejected.

The characteristics of free fall include:

• Constant acceleration due to gravity.
• Velocity changes uniformly over time.
• The object in free fall follows a parabolic path when projected vertically.

By understanding free fall, we learn that once the boulder leaves the volcano, it experiences acceleration only due to gravity. This is crucial for predicting the boulder's motion path and understanding its velocity at different stages.

Acceleration due to Gravity

Acceleration due to gravity is a constant force that pulls objects toward the center of the Earth. Its value is approximately \( -9.8 \, \text{m/s}^2 \). This acceleration affects the boulder's speed and direction during its flight.

Here are some important points about gravitational acceleration:

• It is always directed downward.
• It is independent of the object's direction of motion. Thus, whether the boulder moves upwards, downwards, or is at the peak, the acceleration is the same.
• This uniform acceleration simplifies calculations and lets us predict the boulder's motion accurately.

In projectile motion exercises like this, knowing the acceleration due to gravity helps us solve for various parameters such as time to reach the peak or total flight duration. Despite the boulder's changing velocity, gravitational acceleration remains a consistent factor.