A 65-kg person jumps from the first floor window of a burning building and lands

Question

A 65-kg person jumps from the first floor window of a burning building and lands almost vertically on the ground with a horizontal velocity of 3 m/s and vertical velocity of 9 m/s. Upon impact with the ground he is brought to rest in a short time. The force experienced by his feet depends on whether he keeps his knees stiff or bends them. Find the force on his feet in each case. (a) First find the impulse on the person from the impact on the ground. Calculate both its magnitude and direction. (b) Find the average force on the feet if the person keeps his leg stiff and straight and his center of mass drops by only 1 cm vertically and 1 cm horizontally during the impact. (c) Find the average force on the feet if the person bends his legs throughout the impact so that his center of mass drops by 50 cm vertically and 5 cm horizontally during the impact. (d) Compare the results of part (b) and (c), and draw conclusions about which way is better. You will need to find the time the impact lasts by making reasonable assumptions about the deceleration. Although the force is not constant during the impact, working with constant average force for this problem is acceptable.

Explanation / Answer

a)

Impulse = change in momentum = m*(0 - v) = -mv

where m = 65 kg,

v = initial speed = sqrt(3^2 + 9^2) = 9.49 m/s

So, Impulse = -65*9.49

= -616.9 kg.m/s

b)

using the equation of motion, v^2 = u^2 + 2as

So, in vertical direction, 0^2 = 9^2 + 2*ay*0.01

So, ay = -4050 m/s2

Also In horizontal direction,

0^2 = 3^2 + 2*ax*0.01

So, ax = -450 m/s2

So, average acceleration, a = sqrt(4050^2 + 450^2)

= 4074.9 m/s2

So, average force = m*a = 65*4074.9

= 264868.5 N <------- answer

c)

Similalrly, in vertical direction:

0^2 = 9^2 + 2*ay*0.5

So, ay = -81 m/s2

For thorizontal direction,

0^2 = 3^2 + 2*ax*0.05

So, ax = -90 m/s2

So, a = sqrt(81^2 + 90^2)

= 121.1 m/s2

So, F = m*a = 65*121.1 = 7871.5 N

c)

As the average decelerating force in case 2 is way smaller than that in case 1, it is much better

In case a). time required, t = v/a = sqrt(3^2 + 9^2)/4074.9

= 0.0023 s

In case b) ,time required = sqrt(3^2 + 9^2)/121.1 = 0.078 s

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