A bungee jumper runs off a platform and falls 40m before his bungee cord begins to stretch. Once the bungee cord begins to stretch his velocity slows by 20 m/s in 12m. If the mass of the bungee jumper is 55kg, what is the average force the bungee applies on him?
fell 40 m
v then is
Vi =sqrt(2gh)= sqrt(2*9.81*40)
= 28 m/s
so in the 12 m he slows from
28 to 8 m/s
average speed = (28+8)/2 =36/2 = 18m/s
t = 12 / 18 = 2/3 second
average total force = change in momentum/time
= 55 * 20 *3/2 = 550*3 = 1650 but
add the weight which the bunge also supplies
1650 + m g = 1650+55*9.81
= 2190 N
To solve this problem, we can start by calculating the change in potential energy of the bungee jumper as they fall 40m. The formula for potential energy is:
Potential Energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
Since the bungee jumper is falling, the potential energy is converted into kinetic energy. Therefore, we can equate the change in potential energy to the change in kinetic energy:
Change in Potential Energy = Change in Kinetic Energy
Now, let's calculate the initial potential energy:
Initial Potential Energy = mass * g * initial height
where g is the acceleration due to gravity (approximately 9.8 m/s²) and the initial height is 40m.
Initial Potential Energy = 55 kg * 9.8 m/s² * 40 m
Next, let's calculate the final kinetic energy:
Final Kinetic Energy = 0.5 * mass * (final velocity)²
The final velocity can be determined using the information provided. Since the velocity slows down by 20 m/s in 12 m, we can calculate the final velocity:
Final Velocity = Initial Velocity - (change in velocity)
The change in velocity is given as -20 m/s and the distance traveled during the deceleration is 12m. Assume the initial velocity is zero as the jumper is just starting their fall.
Final Velocity = 0 m/s - (-20 m/s) = 20 m/s
Now, let's calculate the final kinetic energy:
Final Kinetic Energy = 0.5 * 55 kg * (20 m/s)²
Next, we equate the change in potential energy to the change in kinetic energy:
Change in Potential Energy = Change in Kinetic Energy
Initial Potential Energy - Final Potential Energy = Final Kinetic Energy - Initial Kinetic Energy
55 kg * 9.8 m/s² * 40 m - 55 kg * 9.8 m/s² * 12 m = 0.5 * 55 kg * (20 m/s)² - 0
Simplifying this equation:
55 kg * 9.8 m/s² * 28 m = 0.5 * 55 kg * (20 m/s)²
Now, let's solve for the average force applied by the bungee cord. We know that force (F) is equal to the change in momentum divided by the change in time:
Force = (final momentum - initial momentum) / Time
Since there is no change in momentum during the fall (initial momentum is zero), the force can be calculated using the average power formula:
Force = Work / Distance
Remembering that Work = Force * Distance, we can rewrite the equation:
Force = Work / Distance = (Potential Energy - Kinetic Energy) / Distance
Substituting the values we calculated earlier:
Force = (55 kg * 9.8 m/s² * 28 m) / 12 m
Calculating this expression:
Force = 539.4 N
Therefore, the average force applied by the bungee cord on the jumper is 539.4 Newtons.
To find the average force applied by the bungee cord on the jumper, we need to calculate the change in momentum experienced by the jumper due to the slowing down caused by the bungee cord.
To do this, we first need to find the initial velocity of the jumper. We know that the jumper falls freely for 40m before the bungee cord starts to stretch. Assuming no air resistance, we can use the kinematic equation:
v^2 = u^2 + 2as
where v is the final velocity (0 m/s as the jumper stops falling), u is the initial velocity, a is the acceleration due to gravity (-9.8 m/s^2), and s is the vertical distance fallen (40 m).
Plugging in the given values, we have:
(0 m/s)^2 = u^2 + 2 * (-9.8 m/s^2) * 40 m
Simplifying the equation, we get:
0 = u^2 - 784
Rearranging the equation, we find:
u^2 = 784
Taking the square root of both sides, we get:
u = ±28 m/s
Since the jumper is initially falling downward, the negative root is discarded. Therefore, the initial velocity of the jumper is u = 28 m/s.
Next, we can calculate the change in velocity experienced by the jumper due to the bungee cord. The velocity slows down by 20 m/s over a distance of 12 m. We can use the equation:
a = (v - u)/t
where a is the acceleration, v is the final velocity (0 m/s), u is the initial velocity (28 m/s), and t is the time interval (which we need to calculate).
Rearranging the equation, we have:
t = (v - u)/a
Substituting the given values, we get:
t = (0 m/s - 28 m/s)/(-20 m/s)
Simplifying, we find:
t = 1.4 s
Now that we have the time interval, we can calculate the acceleration experienced by the jumper. Recall that acceleration is the change in velocity divided by the time interval:
a = (v - u)/t
Substituting the values, we have:
a = (0 m/s - 28 m/s)/(1.4 s)
Simplifying, we get:
a = -20 m/s^2
The negative sign indicates that the acceleration is in the opposite direction to the initial velocity of the jumper.
Finally, to find the average force applied by the bungee cord on the jumper, we can use Newton's second law:
F = ma
where F is the force, m is the mass of the jumper (55 kg), and a is the acceleration.
Substituting the given values, we have:
F = (55 kg)(-20 m/s^2)
Simplifying, we find:
F = -1100 N
The negative sign indicates that the force is acting in the opposite direction to the initial motion of the jumper. Therefore, the average force applied by the bungee cord on the jumper is 1100 N.