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Chapter 2: Problem 31

(a) If a flea can jump straight up to a height of \(0.440 \mathrm{~m}\), what is its initial speed as it leaves the ground? (b) How long is it in the air?

Short Answer

Expert verified
The initial speed of the flea as it leaves the ground is around \(2.94 \mathrm{~m/s}\). The flea is in the air for about \(0.60 \mathrm{~s}\).

Step by step solution

01

Compute the initial speed of the flea

First, apply the second equation of motion which states: \[v_f^2 = v_i^2 + 2a d,\]with \(v_f\) as the final velocity, \(v_i\) as the initial velocity, \(a\) as acceleration, and \(d\) as displacement. Here, when the flea hits the peak of the jump, it momentarily stops, so \(v_f = 0\), \(a\) is the acceleration due to gravity which is \(-9.81 \mathrm{~m/s}^2\), \(d\) is the height reached by the flea \(0.440 \mathrm{~m}\). So, \[0 = v_i^2 + 2(-9.81)(0.440)\].Solving for \(v_i\) will give the initial speed.
02

Calculate the time the flea is in the air

Next, apply the first equation of motion which states:\[ v_f = v_i + a t,\]where \(v_f\) is the final velocity, \(v_i\) is the initial velocity, \(a\) is acceleration, and \(t\) is time.When the flea lands after the jump, its final velocity is equal but opposite to the initial velocity, \(v_f = -v_i\). Here, \(a\) is the acceleration due to gravity which is \(-9.81 \mathrm{~m/s}^2\).So,\[ -v_i = v_i + (-9.81)t.\]Solving for \(t\) will give the time the flea is in the air.

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Key Concepts

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

Equations of Motion
Understanding the equations of motion is crucial for analyzing objects' movement in physics. These equations enable us to predict an object's position or state of motion at a given time, provided we have relevant initial conditions and constant acceleration. One of the most commonly used equations of motion is \[ v_f^2 = v_i^2 + 2a d, \] where:\
    \
  • \(v_f\) is the final velocity,\
    • \
  • \(v_i\) is the initial velocity,\
    • \
  • \(a\) is the constant acceleration,\
    • \
  • \(d\) is the displacement.\
    • \
\
In the context of a flea's jump, as it reaches the peak height, the final velocity becomes zero, which simplifies our equation and allows us to solve for the initial speed or velocity.
Initial Velocity
Initial velocity, denoted by \(v_i\), is the speed at which an object starts its journey. When a flea jumps up, it leaves the ground with an initial velocity that can be calculated using the appropriate motion equation. In our problem, we are provided with the displacement (the maximum height reached by the flea), and we know the acceleration due to gravity. By setting the final velocity to zero at the peak height and rearranging the equation of motion, we can solve for the initial velocity, which gives us insight into how fast the flea was moving when it left the ground.
Acceleration Due to Gravity
The acceleration due to gravity, typically represented by \(g\), is a constant value of approximately \(-9.81 \mathrm{~m/s}^2\) on Earth, acting downwards. It causes any freely falling object to increase its velocity downward. In kinematics problems involving vertical movement, such as our flea's jump, this acceleration directly influences both the initial velocity necessary for an object to reach a certain height and the time it takes for the object to return to its initial position. The negative sign indicates the direction of acceleration is opposite to the upward direction of the initial jump.
Time of Flight
Time of flight refers to the total duration an object remains airborne. In the example of the flea jumping, the time of flight can be determined once we know the initial velocity and acceleration due to gravity. For an object projected vertically upwards, the time of flight is the time from launch to the moment it returns to the starting point. Calculating it uses the motion equation \[ v_f = v_i + a t, \] where we set the final velocity equal but opposite to the initial velocity when the object hits the ground again. By knowing the time of flight, we get a comprehensive view of how long the flea stays in the air during its jump.

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Most popular questions from this chapter

A small rock is thrown vertically upward with a speed of \(22.0 \mathrm{~m} / \mathrm{s}\) from the edge of the roof of a \(30.0-\mathrm{m}-\) tall building. The rock doesn't hit the building on its way back down and lands on the street below. Ignore air resistance. (a) What is the speed of the rock just before it hits the street? (b) How much time elapses from when the rock is thrown until it hits the street?

A ball is thrown straight up from the ground with speed \(v_{0}\). At the same instant, a second ball is dropped from rest from a height \(H\) directly above the point where the first ball was thrown upward. There is no air resistance. (a) Find the time at which the two balls collide. (b) Find the value of \(H\) in terms of \(v_{0}\) and \(g\) such that at the instant when the balls collide, the first ball is at the highest point of its motion.

It has been suggested, and not facetiously, that life might have originated on Mars and been carried to the earth when a meteor hit Mars and blasted pieces of rock (perhaps containing primitive life) free of the Martian surface. Astronomers know that many Martian rocks have come to the earth this way. (For instance, search the Internet for "ALH 84001 ." ) One objection to this idea is that microbes would have had to undergo an enormous lethal acceleration during the impact. Let us investigate how large such an acceleration might be. To escape Mars, rock fragments would have to reach its escape velocity of \(5.0 \mathrm{~km} / \mathrm{s},\) and that would most likely happen over a distance of about \(4.0 \mathrm{~m}\) during the meteor impact. (a) What would be the acceleration (in \(\mathrm{m} / \mathrm{s}^{2}\) and \(g^{\prime}\) s) of such a rock fragment, if the acceleration is constant? (b) How long would this acceleration last? (c) In tests, scientists have found that over \(40 \%\) of Bacillus subtilis bacteria survived after an acceleration of \(450.000 \mathrm{~g} .\) In light of your answer to part (a), can we rule out the hypothesis that life might have been blasted from Mars to the earth?

In an experiment, a shearwater (a seabird) was taken from its nest, flown \(5150 \mathrm{~km}\) away, and released. The bird found its way back to its nest 13.5 days after release. If we place the origin at the nest and extend the \(+x\) -axis to the release point, what was the bird's average velocity in \(\mathrm{m} / \mathrm{s}\) (a) for the return flight and (b) for the whole episode, from leaving the nest to returning?

A car is stopped at a traffic light. It then travels along a straight road such that its distance from the light is given by \(x(t)=b t^{2}-c t^{3},\) where \(b=2.40 \mathrm{~m} / \mathrm{s}^{2}\) and \(c=0.120 \mathrm{~m} / \mathrm{s}^{3}\). (a) Calculate the average velocity of the car for the time interval \(t=0\) to \(t=10.0 \mathrm{~s}\). (b) Calculate the instantaneous velocity of the car at \(t=0, t=5.0 \mathrm{~s},\) and \(t=10.0 \mathrm{~s} .\) (c) How long after starting from rest is the car again at rest?
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