Problem 46 - College Physics

Question

(a) If a flea can jump straight up to a height of \(22.0 \mathrm{~cm}\), what is its initial speed (in \(\mathrm{m} / \mathrm{s}\) ) as it leaves the ground, neglecting air resistance? (b) How long is it in the air? (c) What are the magnitude and direction of its acceleration while it is (i) moving upward? (ii) moving downward? (iii) at the highest point?

Step-by-Step Solution

Answer

(a) Initial speed is \(2.08\,\text{m/s}\). (b) Total time in air is \(0.424\,\text{s}\). (c) Acceleration is \(9.8\,\text{m/s}^2\) downward in all phases.

Step 1: Understand the Problem

We need to find three things: the initial speed with which the flea leaves the ground, the total time it is in the air, and the acceleration during different phases of its motion. We'll neglect air resistance in our calculations.

Step 2: Identify Known Values

The flea's maximum jump height is 22.0 cm. We'll convert this to meters: \[ 22.0 \text{ cm} = 0.22 \text{ m}. \] The acceleration due to gravity is \( g = 9.8 \text{ m/s}^2 \).

Step 3: Using Kinematic Equation for Initial Speed

We'll use the equation \( v^2 = u^2 + 2as \), where \( v = 0 \text{ m/s} \) at the highest point, \( a = -9.8 \text{ m/s}^2 \), and \( s = 0.22 \text{ m} \).\[ 0 = u^2 - 2 imes 9.8 imes 0.22 \]Solve for \( u \): \[ u = \sqrt{2 \times 9.8 \times 0.22} \approx 2.08 \text{ m/s}. \]

Step 4: Calculate Time to Reach Highest Point

Use the formula \( v = u + at \), with \( v = 0 \text{ m/s} \) at the highest point.\[ 0 = 2.08 - 9.8t \]Solve for \( t \):\[ t \approx \frac{2.08}{9.8} \approx 0.212 \text{ s} \].

Step 5: Determine Total Time in Air

The total time in air is twice the time to reach the highest point (up and down).\[ t_{\text{total}} = 2 \times 0.212 \approx 0.424 \text{ s}. \]

Step 6: Analyze Acceleration at Different Phases

(i) Moving Upward: The magnitude of acceleration is \( 9.8 \text{ m/s}^2 \) downward.(ii) Moving Downward: The magnitude of acceleration is \( 9.8 \text{ m/s}^2 \) downward.(iii) At the Highest Point: The acceleration remains \( 9.8 \text{ m/s}^2 \) downward, as gravity is always acting downward.

Key Concepts

Kinematic Equations

Kinematic equations are important tools that help us understand projectile motion—like our flea jumping problem. These equations relate several key motion parameters: displacement, initial velocity, final velocity, acceleration, and time. When dealing with vertically projected objects, we often use these equations to calculate unknowns, leveraging known values.

• For an object starting with an initial velocity and reaching a certain height, the equation \( v^2 = u^2 + 2as \) is useful. Here, \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is acceleration, and \( s \) is displacement.
• Another key equation is \( v = u + at \), used to find time or final velocity, knowing initial velocity, acceleration, and time.

Understanding these equations is crucial for solving motion problems and predicting how objects move under certain forces.

Acceleration due to Gravity

Acceleration due to gravity is a constant force acting upon objects near the earth’s surface. It is responsible for giving objects a uniform acceleration as they fall. For Earth, this acceleration is \( 9.8 \text{ m/s}^2 \) directed downwards.

• This constant force affects all objects equally, regardless of their size or weight. Hence, our flea and a heavy ball would accelerate similarly if dropped from the same height, assuming no air resistance.
• In projectile motion, gravity is the force that initially decelerates the upward motion until it stops momentarily at the peak, then accelerates it downwards on the return.

Recognizing gravity's role allows us to accurately calculate the different forces acting on our flea during its jump.

Initial Velocity

Initial velocity is a critical factor when analyzing any projectile motion, such as a flea's jump. This refers to the velocity with which an object begins its motion along a defined path. It sets the stage for how high or how far the object will travel.

• The initial velocity determines the object's trajectory. In our exercise, the initial speed is essential to calculate how high the flea can jump.
• Knowing the initial velocity allows us to use kinematic equations effectively to find other variables like maximum height and total time in the air.

In practical terms, if the flea’s initial velocity is higher, it will reach a greater height and stay in the air longer.

Free Fall

Free fall describes the motion of an object under the influence of gravity alone, without any other forces acting, like air resistance. For our flea, its upward and downward motion can be viewed as free fall in both directions after an initial propulsion.

• In true free fall, only gravitational forces act, making calculations with the constant gravitational acceleration straightforward.
• While moving upwards, the flea decelerates to a stop at its peak due to gravity. Then, it transitions smoothly into a downward acceleration, all while maintaining the same gravitational force acting on it.

Understanding free fall assists in predicting the timing and dynamics of the flea's complete jump.