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Chapter 2: Problem 30

When jumping, a flea rapidly extends its legs, reaching a takeoff speed of \(1.0 \mathrm{m} / \mathrm{s}\) over a distance of \(0.50 \mathrm{mm}\). a. What is the flea's acceleration as it extends its legs? b. How long does it take the flea to leave the ground after it begins pushing off?

Short Answer

Expert verified
a. The flea's acceleration as it extends its legs is \(1000 \, m/s^2\). b. It takes the flea 0.001 s to leave the ground after it begins pushing off.

Step by step solution

01

Find the Acceleration

The acceleration of the flea can be derived from the basic equation of motion, which is \(v^2 = u^2 + 2as\), where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(s\) is the distance covered. As the flea starts from rest, \(u = 0\). Substituting the given values into this equation gives \(1.0^2 = 0^2 + 2a*0.0005\). Solving this equation for \(a\) gives \(a = 1.0^2 / (2*0.0005)\), so \(a = 1000 \, m/s^2\).
02

Find the Time

The time it takes the flea to leave the ground can be found using the equation \(v = u + at\), where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time. As the flea starts from rest, \(u = 0\). Substituting the given values into this equation gives \(1.0 = 0 + 1000*t\). Solving this equation for \(t\) gives \(t = 1.0 / 1000\), so \(t = 0.001s\).

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Key Concepts

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

Acceleration
Acceleration is a measure of how quickly an object changes its speed. It is a vector quantity, meaning it has both magnitude and direction. In this exercise, we can see how acceleration allows a flea to jump to its impressive takeoff speed over an incredibly short distance. Acceleration can be calculated using the formula: \[v^2 = u^2 + 2as\] where 鈽� \(v\) is the final velocity, 鈽� \(u\) is the initial velocity, 鈽� \(a\) is the acceleration, and 鈽� \(s\) is the distance. In the context of the flea, since it begins its motion from rest, the initial velocity \(u\) is zero. Therefore, the equation simplifies to \[v^2 = 2as\]. By plugging in the given values, we can accurately compute the acceleration experienced by the flea, which results in an astonishing \(1000 \, \mathrm{m/s^2}\). This high acceleration is what gives the flea its jumping prowess.
Equations of Motion
Equations of motion are essential tools in kinematics that describe the motion of objects. They allow us to predict future positions and velocities of moving objects based on their initial conditions. The primary equations include:
  • \(v = u + at\)
  • \(s = ut + \frac{1}{2}at^2\)
  • \(v^2 = u^2 + 2as\)
Each equation interrelates different variables of motion: displacement \(s\), initial velocity \(u\), final velocity \(v\), acceleration \(a\), and time \(t\). In our flea problem, we used these equations to determine both acceleration and the time taken for the flea to leave the ground. This highlights the versatility of equations of motion, as they can address various aspects of an object's journey from launch to rest.
Velocity
Velocity is a fundamental concept in motion, defined as the rate at which an object changes its position. Unlike speed, velocity also considers the direction of motion, making it a vector quantity. In the exercise, the flea reaches a specific takeoff speed of \(1.0 \, \mathrm{m/s}\), representing its velocity as it jumps. To understand how quickly the flea can achieve this speed, we examine how the velocity changes over time. This is where the foundational equation \(v = u + at\) comes into play. As we calculated, the flea adopts this velocity from rest (\(u = 0\)) in merely \(0.001 \, \mathrm{s}\) due to the previously determined high acceleration of \(1000 \, \mathrm{m/s^2}\). Hence, velocity not only speaks to the speed of an object but also to how quickly that speed is attained or modified.

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Most popular questions from this chapter

In an action movie, the villain is rescued from the ocean by grabbing onto the ladder hanging from a helicopter. He is so intent on gripping the ladder that he lets go of his briefcase of counterfeit money when he is \(130 \mathrm{m}\) above the water. If the briefcase hits the water \(6.0 \mathrm{s}\) later, what was the speed at which the helicopter was ascending?

A football is kicked straight up into the air; it hits the ground 5.2 s later. a. What was the greatest height reached by the ball? Assume it is kicked from ground level. b. With what speed did it leave the kicker"s foot?

A rock climber stands on top of a \(50-\mathrm{m}\) -high cliff overhanging a pool of water. He throws two stones vertically downward \(1.0 \mathrm{s}\) apart and observes that they cause a single splash. The initial speed of the first stone was \(2.0 \mathrm{m} / \mathrm{s}\). a. How long after the release of the first stone does the second stone hit the water? b. What was the initial speed of the second stone? c. What is the speed of each stone as it hits the water?

You are driving to the grocery store at \(20 \mathrm{m} / \mathrm{s}\). You are 110 \(\mathrm{m}\) from an intersection when the traffic light turns red. Assume that your reaction time is \(0.70 \mathrm{s}\) and that your car brakes with constant acceleration. a. How far are you from the intersection when you begin to apply the brakes? b. What acceleration will bring you to rest right at the intersection? c. How long does it take you to stop?

In a 5000 m race, the athletes run \(12 \frac{1}{2}\) laps; each lap is \(400 \mathrm{m}\). Kara runs the race at a constant pace and finishes in 17.5 min. Hannah runs the race in a blistering \(15.3 \mathrm{min},\) so fast that she actually passes Kara during the race. How many laps has Hannah run when she passes Kara?
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