When jumping straight up from a crouched position, an average person can reach a maximum height of about 6060 cm. During the jump, the person's body from the knees up typically rises a distance of around 5050 cm. To keep the calculations simple and yet get a reasonable result, assume that the entire body rises this much during the jump. With what initial speed does the person leave the ground to reach a height of 6060 cm?

Ch 05: Applying Newton's Laws

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 05: Applying Newton's Laws

문제 21a

5장, 문제 21a

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. With what initial speed does the person leave the ground to reach a height of $60$ cm?

검증된 단계별 안내

Step 1: Identify the known values and the equation to use. The maximum height reached by the person is 60 cm, which can be converted to meters: $ h = 0.60 \, \text{m} $. The acceleration due to gravity is $ g = 9.8 \, \text{m/s}^2 $. The initial velocity $ v_0 $ is what we need to find. Use the kinematic equation for vertical motion: $ v^2 = v_0^2 - 2gh $, where $ v $ is the final velocity at the maximum height ($ v = 0 \, \text{m/s} $).

Step 2: Rearrange the kinematic equation to solve for $ v_0 $. Since $ v = 0 $ at the maximum height, the equation simplifies to $ v_0^2 = 2gh $. Taking the square root of both sides gives $ v_0 = \sqrt{2gh} $.

Step 3: Substitute the known values into the equation. Replace $ g $ with $ 9.8 \, \text{m/s}^2 $ and $ h $ with $ 0.60 \, \text{m} $. The equation becomes $ v_0 = \sqrt{2 \cdot 9.8 \cdot 0.60} $.

Step 4: Perform the calculation inside the square root. Multiply $ 2 \cdot 9.8 \cdot 0.60 $ to find the value under the square root.

Step 5: Take the square root of the result from Step 4 to find the initial speed $ v_0 $. This is the speed at which the person leaves the ground to reach the maximum height of 60 cm.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this context, kinematics helps us understand how the initial speed of the jumper relates to the maximum height achieved during the jump.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It is calculated using the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is height. This concept is crucial for determining the initial speed required to reach a specific height, as the potential energy at the peak of the jump must equal the kinetic energy at takeoff.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In the context of the jump, the kinetic energy of the person at takeoff is converted into gravitational potential energy at the peak height. This relationship allows us to calculate the initial speed needed to achieve the desired height by equating the kinetic and potential energies.