Starting from rest, a 2.4x10-4 kg flea springs straight upward. While the flea is pushing off from the ground, the ground exerts an average upward force of 0.42 N on it. This force does 1.5x10-4 J of work on the flea. (a) What is the flea's speed when it leaves the ground? (b) How far upward does the flea move while it is pushing off? Ignore both air resistance and the flea's weight
To solve this problem, we need to use the work-energy principle, which states that the work done on an object is equal to its change in kinetic energy. We can break down the problem into two parts:
(a) Calculating the flea's speed when it leaves the ground:
The work done on the flea by the ground is given as 1.5x10^(-4) J. We can equate this to the change in kinetic energy of the flea.
Work = Change in Kinetic Energy
1.5x10^(-4) J = (1/2)mv^2
Where m is the mass of the flea and v is its final velocity.
Rearranging the equation, we get:
v^2 = (2 * 1.5x10^(-4) J) / m
v^2 = 3.0x10^(-4) J / (2.4x10^(-4) kg)
v^2 = 1.25 J/kg
Taking the square root of both sides, we find:
v = √(1.25 J/kg)
Calculating this, we get:
v ≈ 1.12 m/s
Therefore, the flea's speed when it leaves the ground is approximately 1.12 m/s.
(b) Calculating the distance the flea moves upward while pushing off:
To calculate the distance traveled, we need to determine the displacement of the flea during the time it is pushing off.
The work done is given as 0.42 N, which is the product of the average upward force and the distance traveled (d).
Work = Force * Distance
0.42 N * d = 1.5x10^(-4) J
Rearranging the equation, we have:
d = 1.5x10^(-4) J / 0.42 N
Calculating this, we get:
d ≈ 3.57x10^(-4) m
Therefore, while pushing off, the flea moves upward approximately 3.57x10^(-4) meters.
To find the flea's speed when it leaves the ground, we can use the work-energy principle. The work done on the flea by the ground is equal to the change in the flea's kinetic energy.
(a) Calculate the kinetic energy of the flea after leaving the ground:
Since the flea starts from rest, its initial kinetic energy is zero. The work done on the flea is equal to the change in kinetic energy:
Work = Change in Kinetic Energy
0.42 N * distance = (1/2) * m * V^2
where m = 2.4x10^-4 kg (mass of the flea) and V is the speed of the flea when it leaves the ground.
Rearranging the equation, we get:
V^2 = (2 * work) / m
Substituting the given values, we get:
V^2 = (2 * 1.5x10^-4 J) / 2.4x10^-4 kg
V^2 = 1.25
Taking the square root of both sides, we find:
V ≈ 1.12 m/s
Therefore, the flea's speed when it leaves the ground is approximately 1.12 m/s.
(b) To find how far upward the flea moves while it is pushing off, we need to determine the distance traveled. We can use the work-energy principle once again:
Work = Change in Potential Energy
0.42 N * distance = m * g * h
where m = 2.4x10^-4 kg (mass of the flea), g = 9.8 m/s^2 (acceleration due to gravity), and h is the distance traveled.
Rearranging the equation, we get:
distance = (m * g * h) / 0.42 N
Substituting the given values, we get:
distance = (2.4x10^-4 kg * 9.8 m/s^2 * 1.5x10^-4 J) / 0.42 N
distance ≈ 0.016 m
Therefore, the flea moves approximately 0.016 meters upward while pushing off from the ground.