A boy weighing 4.2 102 N jumps from a height of 1.80 m to the ground below. Assume that the force of the ground on his feet is constant.
(a) Compute the force of the ground on his feet if he jumps stiff-legged, the ground compresses 2.20 cm, and the compression of tissue and bones is negligible.
(b) Compute the force his legs exert on his upper body (trunk, arms, and head), which weighs 2.50 102 N, under the conditions assumed above.
(c) Now suppose that his knees bend on impact, so that his trunk moves downward 44.0 cm during deceleration. Compute the force his legs exert on his upper body.
To solve parts (a), (b), and (c) of this problem, we'll need to apply the principles of kinematics and Newton's laws of motion.
(a) To compute the force of the ground on the boy's feet if he jumps stiff-legged, we can use the concept of work done by compression. The work done by the force of the ground is equal to the change in potential energy. The formula to calculate work done is given by:
Work = Force x Distance
In this case, the distance is the compression of the ground, which is given as 2.20 cm (or 0.022 m). The gravitational potential energy is equal to the boy's weight times the height from which he jumps, which is (4.2 * 10^2 N) * (1.80 m).
So, the work done by the ground is equal to the change in potential energy:
Work = (4.2 * 10^2 N) * (1.80 m) = 7.56 * 10^2 J
Since work is equal to force multiplied by distance, we can rearrange the formula to solve for force:
Force = Work / Distance
Force = (7.56 * 10^2 J) / (0.022 m) ≈ 3.44 * 10^4 N
Therefore, the force of the ground on the boy's feet when he jumps stiff-legged is approximately 3.44 * 10^4 N.
(b) To compute the force his legs exert on his upper body under the given conditions, we can use Newton's third law of motion, which states that for every action, there is an equal and opposite reaction.
The force his legs exert on his upper body is equal to the force of the ground on his feet, since they are connected and facing in opposite directions. Therefore, the force his legs exert on his upper body is also approximately 3.44 * 10^4 N.
(c) In this case, since the boy's knees bend on impact, his trunk moves downward during deceleration by 44.0 cm (or 0.44 m). We need to calculate the change in potential energy and use it to find the force exerted by his legs on his upper body.
The change in potential energy is given by the weight of his upper body multiplied by the change in height:
Change in potential energy = (2.50 * 10^2 N) * (0.44 m)
Since work equals force multiplied by distance, the work done by the boy's legs is equal to the change in potential energy. Therefore, the force his legs exert on his upper body can be calculated using the formula:
Force = Change in potential energy / Distance
Force = ((2.50 * 10^2 N) * (0.44 m)) / (0.44 m) = 2.50 * 10^2 N
Hence, the force his legs exert on his upper body when his knees bend on impact is 2.50 * 10^2 N.
In summary:
(a) The force of the ground on his feet when he jumps stiff-legged is approximately 3.44 * 10^4 N.
(b) The force his legs exert on his upper body under the given conditions is approximately 3.44 * 10^4 N.
(c) The force his legs exert on his upper body when his knees bend on impact is 2.50 * 10^2 N.