A pole-vaulter just clears the bar at 4.07 m and falls back to the ground. The change in the vaulter's potential energy during the fall is -3900 J. What is his weight?
PEi + KEi = PEf + KEf
mgh + (0) = (0) + (1/2)mv^2
mgh = (1/2)mv^2
(9.8)(4.07)m = 3900
m = 97.8 kg
Fg = mg = (9.8)(97.8)
Fg = 958 N
Well, I hope the pole-vaulter didn't fall too hard! Now, to find his weight, we can use the formula for gravitational potential energy:
Potential energy = weight * height
Since the change in potential energy is -3900 J, and the height is 4.07 m, we can rearrange the formula to solve for weight:
Weight = Change in potential energy / height
Weight = -3900 J / 4.07 m
Hmm, it seems like the negative sign here indicates that something's amiss. I must be experiencing some technical difficulties trying to calculate a vaulter with negative weight. Perhaps there's an error in the given information or the way I'm interpreting it. Let's try again, shall we?
To determine the vaulter's weight, we need to first calculate the height from which the vaulter falls.
Given:
Change in potential energy (ΔPE) = -3900 J
Acceleration due to gravity (g) = 9.8 m/s^2
Potential energy (PE) is given by the formula PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height.
Since the vaulter is falling, the change in potential energy (ΔPE) can be written as:
ΔPE = PE_final - PE_initial.
Here, the initial potential energy (PE_initial) is zero, as it is at the ground level.
So, ΔPE = PE_final - 0.
Therefore, ΔPE = PE_final.
ΔPE = mgh.
mgh = -3900 J.
Now, let's solve for h (height):
h = -3900 J / (mg).
The weight of the vaulter is given by the formula W = mg.
Now, let's plug in the given values:
h = -3900 J / (m x 9.8 m/s^2).
h = -3900 J / (9.8 m/s^2 x m).
Since the unit for height is meters, and the unit for potential energy is Joules, the negative sign indicates that the fall is in the downward direction.
Now, let's rearrange the equation to solve for m (mass):
h x 9.8 m/s^2 x m = -3900 J.
m^2 = -3900 J / (9.8 m/s^2).
m^2 = -3900 kg·m^2/s^2.
Taking the square root of both sides, we obtain:
m = sqrt(-3900 kg·m^2/s^2).
Since the value under the square root is negative, the given information is inconsistent. The weight cannot be determined without additional information or if a mistake has been made in the calculations.
To determine the weight of the pole-vaulter, we can use the formula for potential energy, which is given by:
Potential Energy = Weight × Height × g
where:
Potential Energy is the change in potential energy during the fall (-3900 J),
Weight is the weight of the pole-vaulter, and
Height is the height the pole-vaulter fell from (4.07 m).
g is the acceleration due to gravity (approximately 9.8 m/s^2).
We can rearrange the formula to solve for weight:
Weight = Potential Energy / (Height × g)
Substituting the given values:
Weight = -3900 J / (4.07 m × 9.8 m/s^2)
Now we can calculate the weight using a calculator:
Weight ≈ -3900 J / (39.886 m²/s²)
Weight ≈ -97.773 N
Since weight is a downward force, the negative sign indicates that the weight is acting in the opposite direction to the positive direction. Therefore, the weight of the pole-vaulter is approximately 97.773 N.