91Ó°ÊÓ

In a physics lab experiment, you release a small steel ball at various heights above the ground and measure the ball's speed just before it strikes the ground. You plot your data on a graph that has the release height (in meters) on the vertical axis and the square of the final speed (in m\(^2\)/s\(^2\)) on the horizontal axis. In this graph your data points lie close to a straight line. (a) Using \(g\) = 9.80 m/s\(^2\) and ignoring the effect of air resistance, what is the numerical value of the slope of this straight line? (Include the correct units.) The presence of air resistance reduces the magnitude of the downward acceleration, and the effect of air resistance increases as the speed of the object increases. You repeat the experiment, but this time with a tennis ball as the object being dropped. Air resistance now has a noticeable effect on the data. (b) Is the final speed for a given release height higher than, lower than, or the same as when you ignored air resistance? (c) Is the graph of the release height versus the square of the final speed still a straight line? Sketch the qualitative shape of the graph when air resistance is present.

Step by step solution

01

Understanding the Relationship

The physics of freefall without air resistance follows the equation for motion: \( v^2 = u^2 + 2gh \). For an object released from rest, \( u = 0 \), making it \( v^2 = 2gh \).

02

Identifying the Slope

From the relation \( v^2 = 2gh \), the variable \( h \) (height) corresponds to the dependent variable, graphed on the vertical axis, as \( h = \frac{v^2}{2g} \). The form shows that the plot \( h \) vs. \( v^2 \) is a straight line with slope \( \frac{1}{2g} \). Thus, slope = \( \frac{1}{2 \times 9.8} = 0.051 \) with units meters per \( (m^2/s^2) \).

03

Determine the Effect of Air Resistance

With air resistance, the actual acceleration is less than the gravitational acceleration, leading the speed just before impact to be slower. Therefore, for a given height, the final speed is lower than when air resistance is ignored.

04

Analyze the New Graph Shape

With air resistance, the relationship between \( h \) and \( v^2 \) becomes nonlinear; the path curves and becomes concave downwards instead of a straight line, as greater height increases cause less steep increases in \( v^2 \).

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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 describes the motion of objects without considering the causes of this motion. In freefall motion, which is a type of kinematic motion, an object moves only under the influence of gravity, neglecting any resistance.

• When an object is released from rest, its initial velocity (\( u \)) is zero.
• The equation of motion that describes freefall is \( v^2 = u^2 + 2gh \).

Here,

• \( v \) is the final velocity just before the object hits the ground,
• \( h \) is the height from which the object is dropped, and
• \( g \) is the acceleration due to gravity, approximately \( 9.80 \, \text{m/s}^2 \) on Earth.

Since \( u = 0 \) for objects in freefall starting from rest, the equation simplifies to \( v^2 = 2gh \). This relationship tells us that the final speed of a dropped object depends only on the height of the drop and the constant acceleration of gravity.

Air Resistance

Air resistance is a force that opposes the motion of an object through the air. It plays a significant role in determining the object's motion alongside gravity, especially at higher speeds.
When air resistance is considered, the actual downward acceleration of the object becomes less than the gravitational acceleration (\( g \)).

• This results in the object falling slower than it would in a vacuum.
• For the tennis ball experiment described, the effect of air resistance becomes noticeable.

In the presence of air resistance:

• The final speed (\( v \)) for a given height (\( h \)) is lower compared to when air resistance is ignored.
• As speed increases, the force of air resistance also increases, making the object reach a terminal velocity if the descent is prolonged enough.

This change in speed significantly affects the outcomes and interpretation of freefall experiments.

Graphical Analysis

Graphical analysis lets students visualize the relationships between variables in kinematic experiments. In the freefall experiment, plotting release height versus the square of final speed helps reveal the influence of different forces on motion.
When plotting \( h \) against \( v^2 \) without air resistance, the plot yields a straight line because of the direct linear relationship \( h = \frac{v^2}{2g} \).

• The slope here is \( \frac{1}{2g} \).
• It quantifies the proportionality between the variables without external interference like air resistance.

However, once air resistance is included in the experiment:

• The graph becomes nonlinear, and the plot visually shifts to a curve that concaves downward.
• This curvature indicates that higher release heights result in less increase in \( v^2 \), illustrating the slowing effect of air resistance at greater speeds or heights.

Therefore, understanding these graphical shifts is critical in analyzing motion comprehensively under different conditions.