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. 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. Step 1: Identify the forces acting on the jumper during the jump. The ground exerts an upward force (normal force) on the jumper, which must overcome the jumper's weight (gravitational force) and provide the additional force needed to accelerate the jumper upward.
2. Step 2: Use Newton's second law of motion, which states that the net force is equal to the mass times the acceleration: $F=ma$. Here, the net force is the difference between the upward force exerted by the ground and the jumper's weight.
3. Step 3: Calculate the acceleration of the jumper during the jump. Use the kinematic equation: $v{2}^2=u{2}^2+2as$, where $v$ is the final velocity (0 m/s at the peak of the jump), $u$ is the initial velocity, $a$ is the acceleration, and $s$ is the displacement (50 cm or 0.5 m). Rearrange the equation to solve for $a$.
4. Step 4: Relate the jumper's weight to their mass using the equation $w=mg$, where $g$ is the acceleration due to gravity (approximately 9.8 m/s²). Substitute this into the net force equation to express the upward force exerted by the ground in terms of the jumper's weight.
5. Step 5: Combine the equations from Steps 2 and 4 to find the total force exerted by the ground. This force is equal to the jumper's weight plus the additional force required to accelerate the jumper upward. The final expression will be in terms of the jumper's weight $w$.

Key Concepts

Newton's Third Law of Motion

Newton's Third Law states that for every action, there is an equal and opposite reaction. In the context of jumping, when a person exerts a force downward on the ground, the ground exerts an equal force upward on the person. This reaction force is crucial for understanding how the jumper is propelled into the air.

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). In this scenario, the jumper's weight plays a significant role in determining the net force acting on them during the jump, influencing how high they can ascend.

Net Force and Acceleration

The net force acting on an object is the vector sum of all forces acting on it. According to Newton's second law (F = ma), this net force determines the object's acceleration. During the jump, the ground's upward force must overcome the jumper's weight to create a net upward force, allowing the jumper to accelerate upwards.