When jumping straight up from a crouched position, an average person can reach a maximum height of about 60 cm. During the jump, the person's body from the knees up typically rises a distance of around 50 cm. To keep the calculations simple and yet get a reasonable result, assume that the entire body rises this much during the jump. (c) In terms of this jumper's weight w, what force does the ground exert on him or her during the jump?

Verified step by step guidance

1. Identify the initial and final velocities: When jumping, the initial velocity is what you need to find, and the final velocity at the maximum height is 0 m/s because the jumper momentarily stops moving upward.
2. Use the kinematic equation to relate the initial velocity, final velocity, acceleration due to gravity, and the height: \(v_f^2 = v_i^2 + 2a imes d\), where \(v_f\) is the final velocity, \(v_i\) is the initial velocity, \(a\) is the acceleration (gravity, which is -9.8 m/s^2), and \(d\) is the distance risen (0.5 m).
3. Solve the kinematic equation for the initial velocity \(v_i\): \(v_i = \\sqrt{v_f^2 - 2ad}\).
4. Calculate the force exerted by the ground using Newton's second law, which states that the force is equal to mass times acceleration: \(F = m imes a\). Here, the acceleration is not just gravity but the net acceleration during the jump.
5. Determine the net acceleration by considering the change in velocity over time. The time can be estimated using the motion equations, and then use \(F = m imes a\) to find the force, where \(m\) is the mass of the jumper (which can be found using \(w = m imes g\), where \(w\) is the weight and \(g\) is the acceleration due to gravity).

Key Concepts

Newton's Second Law of Motion

Newton's Second Law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration (F = ma). In the context of the jump, this law helps us understand how the force exerted by the ground must counteract both the weight of the jumper and provide the necessary upward acceleration to achieve the jump height.

Weight and Gravitational Force

Weight is the force exerted by gravity on an object, calculated as the product of mass and the acceleration due to gravity (w = mg). For the jumper, this weight acts downward, and understanding this force is crucial for determining the net force exerted by the ground during the jump, as it must overcome this weight to initiate upward motion.

Net Force and Acceleration

The net force is the total force acting on an object after all opposing forces are considered. During the jump, the ground must exert a force greater than the jumper's weight to create a net upward force, resulting in acceleration. This concept is essential for calculating the exact force the ground exerts on the jumper at the moment of takeoff.