Question

In the 1996 Olympic Games, the Bulgarian high jumper Stefka Kostadinova set a women's Olympic record for this event with a jump of \(2.05 \mathrm{~m}\); see Fig. \(12-17\). Other things being equal, how high might she have jumped on the Moon, where the surface gravity is only \(1.67 \mathrm{~m} / \mathrm{s}^{2} ?\) (Hint: The height that "counts" is the vertical distance her center of gravity rose after her feet left the ground. Assume that, at the instant her feet lost contact, her center of gravity was \(110 \mathrm{~cm}\) above ground level. Assume also that, as she clears the bar, her center of gravity is at the same height as the bar.)

Short answer

The answer is that Stefka Kostadinova might have jumped approximately \(11.31 \, m\) high on the Moon.

Step-by-step solution

Identify the problem type

Here, we are dealing with a physics problem that involves free fall under gravity. Specifically, we are asked to compare heights of free fall on Earth and on the Moon.

Calculate the total time for ascent and descent on Earth

The equation for height in a physics motion problem is \( h = h_0 + v_0t - 0.5gt^2 \). However, since on the highest point the velocity is 0, the equation can be rewritten in terms of time \( t = \sqrt{2h/g} \). When considering the time for ascent and descent, the time actually doubles. In our case, the initial height \( h_0 \) is 1.10m and the height gained \( h \) is 0.95m(2.05m - 1.10m). So, for the Earth where gravity \( g=9.8 \, m/s^2 \), the total time becomes \( t = 2\sqrt{2*0.95/9.8} \).

Calculate the total height she might have jumped on the Moon

We can calculate the total height she might have jumped on the Moon by taking into account that ascent and descent times are equal to that of Earth, but gravity is different. Re-writing our equation from step 2 and solve for height: \( h = g*(t^2)/2 \). Putting the calculated value for \( t \) and using moon's gravity \( g=1.67 \, m/s^2 \), we get the height she could have jumped on the Moon avoiding the initial height. To get the total height, we just need to add the initial height which is \( 1.10 \, m \) on Moon too.

Verify the results

Check the results for plausibility by considering that the gravity on the Moon is significantly smaller than that on Earth. Therefore, the height jumped should be significantly higher.

Key concepts

Gravity

Gravity is a fundamental force that pulls objects towards the center of a planet, like Earth or the Moon. It determines how fast objects fall and influences the motion of everything around us. On Earth, gravity accelerates objects at \(9.8 \, \text{m/s}^2\). This means for every second an object is in free fall, its speed increases by about 9.8 meters per second.

Gravity's effects can be observed in everyday life. For a high jumper like Stefka Kostadinova, gravity is what pulls her back to the ground after her leaps. Understanding how gravity works helps us calculate how high a person can jump or how far they can throw a ball. It's all about balance between the initial muscle-powered leap and the gravitational pull bringing her down.

Vertical Displacement

Vertical displacement refers to the change in vertical position of an object or, in this case, the high jumper. It’s the measure of how high she rises above the initial ground level. In Stefka’s jump, when we talk about vertical displacement, we are looking at the difference between where her center of gravity starts and where it ends at the highest point.

Let's break it down: her initial center of gravity was 1.10 meters above the ground, and she reached a peak of 2.05 meters. Thus, the vertical displacement in her jump was \(2.05 \, \text{m} - 1.10 \, \text{m} = 0.95 \, \text{m}\). Vertical displacement helps us understand the efficiency of her jump and forms the basis to compare how high she might jump under different gravitational conditions.

Center of Gravity

The center of gravity is the specific point where the entire weight of an object is evenly distributed and balanced. For a high jumper, it's crucial because it affects how high and far they can jump.

Stefka Kostadinova's center of gravity plays a key role in determining how her body moves through the air. Before the jump, her center of gravity was 1.10 meters from the ground. At her peak, it was at the height of the bar. Understanding the center of gravity allows athletes to optimize their body position to maximize their jump height. Skilled jumpers learn to manipulate their center of gravity to achieve impressive heights.

Moon's Gravity

The Moon's gravity is much weaker than Earth's, at only \(1.67 \, \text{m/s}^2\). This means that objects on the Moon fall slower and can be thrown higher. If Stefka Kostadinova were to jump on the Moon, she could reach greater heights due to this reduced gravitational pull.

The exercise involves calculating how much higher she could jump with this weaker gravity. With Earth's gravity holding her back less, the same force of her jump would allow her center of gravity to go significantly higher. This demonstrates how different gravitational environments affect motion, helping us understand physics not just on Earth, but in space as well.