A pole-vaulter of mass 60.2 kg vaults to a height of 6.1 m before dropping to thick padding placed below to cushion her fall.
(a) Find the speed with which she lands.
(b) If the padding brings her to a stop in a time of 0.52 s, what is the average force on her body due to the padding during that time interval?
The world record for women's pole vault is not 6.1 meters. It is about 5.06 meters.
Whoever is making sure of gender neutrality in your textbook should get his or her facts straight.
http://www.youtube.com/watch?v=8_RfK2rp2To
(a) V = sqrt(2 g H)
(b)Fav*time = Momentum at impact
Solve for average force, Fav.
6.1 m is actually a few cm short of the men's pole vault world record
To find the speed with which the pole-vaulter lands, we can use the principle of conservation of energy. At the highest point of her jump, all of her initial potential energy is converted into kinetic energy.
(a) First, we need to find the potential energy at the highest point. The potential energy is given by the formula:
Potential Energy = mass × gravity × height
Given:
Mass of the pole-vaulter (m) = 60.2 kg
Height reached (h) = 6.1 m
Acceleration due to gravity (g) = 9.8 m/s²
Potential Energy = 60.2 kg × 9.8 m/s² × 6.1 m = 3593.40 Joules
Since all the potential energy is converted into kinetic energy, we can equate the two:
Kinetic Energy = Potential Energy
0.5 × mass × velocity² = 3593.40 Joules
Rearranging the equation, we get:
velocity² = (2 × Potential Energy) / mass
velocity² = (2 × 3593.40 Joules) / 60.2 kg
velocity² = 119.61 m²/s²
Taking the square root of both sides, we find:
velocity = √(119.61 m²/s²) ≈ 10.95 m/s
Therefore, the speed with which the pole-vaulter lands is approximately 10.95 m/s.
(b) To find the average force on her body due to the padding, we can use the equation:
Average Force = change in momentum / time
The change in momentum is given by:
change in momentum = mass × (final velocity - initial velocity)
Given:
Mass of the pole-vaulter (m) = 60.2 kg
Initial velocity (u) = 10.95 m/s (as found in part a)
Final velocity (v) = 0 m/s (she comes to a stop)
Average Force = (mass × (final velocity - initial velocity)) / time
Average Force = (60.2 kg × (0 m/s - 10.95 m/s)) / 0.52 s
Average Force = (-659.9 N) / 0.52 s
Average Force ≈ -1271 N
(Note: The negative sign indicates that the force is directed in the opposite direction of the motion, which is the deceleration force exerted by the padding.)
Therefore, the average force on her body due to the padding during the time interval of 0.52 seconds is approximately 1271 Newtons.
To find the speed with which the pole-vaulter lands, we need to use the conservation of energy equation. We can assume that the initial energy equals the final energy.
(a) Finding the speed:
The initial energy of the pole-vaulter is equal to her potential energy at the highest point of her jump, which is given by the formula:
Initial Energy = m * g * h
where m is the mass of the pole-vaulter, g is the acceleration due to gravity (9.8 m/s²), and h is the height reached (6.1 m).
So, the initial energy is:
Initial Energy = 60.2 kg * 9.8 m/s² * 6.1 m
The final energy is her kinetic energy just before landing and can be calculated using the formula:
Final Energy = (1/2) * m * v²
where v is the final velocity (the speed with which she lands).
Now, equating the initial and final energies, we have:
Initial Energy = Final Energy
m * g * h = (1/2) * m * v²
Simplifying and solving for v, we get:
v = sqrt(2 * g * h)
Substituting the given values, we have:
v = sqrt(2 * 9.8 m/s² * 6.1 m)
v ≈ 8.74 m/s
So, the speed with which the pole-vaulter lands is approximately 8.74 m/s.
(b) Finding the average force:
To find the average force on the pole-vaulter's body due to the padding, we can use the impulse-momentum theorem. The impulse J imparted to an object is equal to the change in its momentum.
The impulse is given by the formula:
Impulse (J) = force (F) * time (t)
The initial momentum of the pole-vaulter is zero (assuming she starts from rest), and the final momentum is given by:
Final Momentum = m * v
where v is the landing speed (8.74 m/s).
According to the impulse-momentum theorem, the impulse J is equal to the change in momentum, given by:
J = Final Momentum - Initial Momentum
Since the initial momentum is zero, the equation simplifies to:
J = Final Momentum
So, the impulse J is equal to the average force exerted by the padding on the pole-vaulter's body for a given time interval.
Substituting the given values, we have:
J = m * v
We know the time t is given as 0.52 s, so:
J = m * v = m * a * t
where a is the acceleration experienced by the pole-vaulter during the time interval.
Simplifying, we get:
F * t = m * a * t
The t (time) cancels out, and we are left with:
F = m * a
Now, let's calculate the average force:
F = 60.2 kg * 8.74 m/s / 0.52 s
F ≈ 1014 N
So, the average force on the pole-vaulter's body due to the padding during that time interval is approximately 1014 Newtons.