Starting from rest, a 2.3x10-4 kg flea springs straight upward. While the flea is pushing off from the ground, the ground exerts an average upward force of 0.44 N on it. This force does 2.7x10-4 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.
For a) I got 1.53 m/s, but I don't know how to solve for b). Please help and explain!
Thanks
initial work=initial KE
2.7e-4=1/2 mass*(v^2) solve for v.
b. max height PE=initial KE
mgh=initial work
solve for height h.
A.
(1/2) m v^2 = 2.7*10^-4 J
(1/2) (2.3*10^-4) v^2 = 2.7*10^-4
v = 1.53 m/s agreed
B.
Work done = force * distance
2.7*10^-4 = 0.44 d
solve for d
I interpreted as rise during push, not total height
Well, I'm glad you've tackled part a already! Let's move on to part b.
To calculate the distance upward that the flea moves while pushing off, we can use the work-energy principle. This principle states that the work done on an object equals the change in its kinetic energy. In this case, the work done by the ground is equal to the change in the flea's kinetic energy while pushing off.
Since the flea starts from rest, its initial kinetic energy is zero. The work done by the ground is given as 2.7x10-4 J. Therefore, this work is equal to the final kinetic energy of the flea.
We can calculate the final kinetic energy of the flea using the formula:
Kinetic energy = (1/2) * mass * velocity^2
The mass of the flea is given as 2.3x10-4 kg, and we need to find the final velocity.
Using algebra, we can rearrange the formula to solve for the final velocity:
velocity = √(2 * kinetic energy / mass)
Plugging in the known values:
velocity = √(2 * 2.7x10-4 J / 2.3x10-4 kg)
Evaluating this expression, we find that the velocity is approximately 1.32 m/s.
Now, to determine the distance traveled by the flea, we need to know the time it took for the flea to leave the ground. However, the time is not given in the problem. Without it, we cannot calculate the distance accurately.
Therefore, without the time information, it is impossible to solve for the distance upward the flea moves while pushing off.
To solve for part (b) of the problem, we need to use the work-energy principle. The work-energy principle states that the work done on an object is equal to the change in its kinetic energy.
Let's start by finding the initial kinetic energy of the flea when it leaves the ground. Since the flea starts from rest, its initial kinetic energy is zero.
Next, we need to find the final kinetic energy of the flea. We can calculate this using the work done on the flea.
Given that the work done on the flea is 2.7x10^(-4) J and the flea's final speed is what we need to solve for, we can use the equation for work:
Work = change in kinetic energy
2.7x10^(-4) J = final kinetic energy - initial kinetic energy
Since the initial kinetic energy is zero, we can simplify the equation to:
2.7x10^(-4) J = final kinetic energy
Now, we need to find the final kinetic energy of the flea. To do this, we can use the equation for kinetic energy:
Kinetic energy = (1/2) * mass * velocity^2
In this case, the mass of the flea is 2.3x10^(-4) kg and we want to solve for the final velocity.
Plugging in the values into the kinetic energy equation, we get:
2.7x10^(-4) J = (1/2) * (2.3x10^(-4) kg) * (velocity)^2
Now, we can solve for the final velocity:
2.7x10^(-4) J = (1.15x10^(-4) kg) * (velocity)^2
Dividing both sides of the equation by (1.15x10^(-4) kg):
(2.7x10^(-4) J) / (1.15x10^(-4) kg) = (velocity)^2
Calculating this value, we get:
velocity^2 = 2.3478 m^2/s^2
Finally, taking the square root of both sides, we find the final velocity:
velocity ≈ 1.534 m/s
So, the speed of the flea when it leaves the ground is approximately 1.534 m/s.
Now, for part (b) of the problem, we need to find how far upward the flea moves while it is pushing off. To solve for this, we need to use the equation of motion for vertical motion:
vf^2 = vi^2 + 2 * a * d
Since the flea starts from rest, its initial velocity is zero, and the final velocity is the velocity we calculated in part (a).
Plugging these values into the equation, we get:
(1.534 m/s)^2 = 0^2 + 2 * a * d
Now, since we know that the average upward force exerted by the ground is 0.44 N, we can calculate the acceleration using Newton's second law (F = ma):
0.44 N = (2.3x10^(-4) kg) * a
Now, solving for the acceleration (a):
a = (0.44 N) / (2.3x10^(-4) kg) ≈ 1913.04 m/s^2
Plugging this value into the equation for vertical motion, along with the final velocity, we get:
(1.534 m/s)^2 = 0^2 + 2 * (1913.04 m/s^2) * d
Simplifying this equation, we find:
2.352356 m^2/s^2 = 3826.08 m/s^2 * d
Finally, solving for the distance (d), we get:
d = 2.352356 m^2/s^2 / (3826.08 m/s^2) ≈ 0.0006144 m
So, the flea moves approximately 0.0006144 meters (or approximately 0.6144 mm) upward while it is pushing off.