Problem 41 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Extreme-sports enthusiasts have been known to jump off the top of El Capitan, a shecr granite cliff of height 910 \(\mathrm{m}\) in Yosemite National Park. Assume a jumper runs horizontally off the top of EI Capitan with speed 5.0 \(\mathrm{m} / \mathrm{s}\) and enjoys a freefall until she is 150 \(\mathrm{m}\) above the valley floor, at which time she opens her parachute (Fig. 41\()\) .(a) How long is the jumper in frecfall? Ignore air resistance. \((b)\) It is important to be as far away from the cliff as possible before opening the parachute. How far from the cliff is this jumper when she opens her chute?

Step-by-Step Solution

Answer

(a) 12.44 s; (b) 62.2 m.

Step 1: Understand the Problem

A jumper runs horizontally off a cliff with a speed of 5.0 m/s, starting from a height of 910 m, and free falls until reaching a height of 150 m. We need to find out how long the jumper is in free fall and how far she travels horizontally during this time. We will ignore air resistance for simplicity.

Step 2: Free Fall Calculation

The height difference between the jump point (910 m) and the parachute opening point (150 m) is the total distance traveled vertically: \[ h = 910 - 150 = 760 \, \mathrm{m} \] Using the kinematic equation for free fall: \[ h = \frac{1}{2} g t^2 \] Where \( g = 9.81 \, \mathrm{m/s^2} \) is the acceleration due to gravity, we solve for \( t \): \[ 760 = \frac{1}{2} \times 9.81 \times t^2 \] Rearranging gives: \[ t^2 = \frac{760}{4.905} \approx 154.84 \] Taking the square root gives: \[ t \approx \sqrt{154.84} \approx 12.44 \, \mathrm{s} \] So the jumper is in free fall for approximately 12.44 seconds.

Step 3: Horizontal Distance Calculation

During free fall, the horizontal distance \( d \) traveled can be calculated using the constant horizontal speed: \[ d = v_{horizontal} \times t \] \[ v_{horizontal} = 5.0 \, \mathrm{m/s} \] And \( t = 12.44 \, \mathrm{s} \) from the earlier step, thus: \[ d = 5.0 \times 12.44 \approx 62.2 \, \mathrm{m} \] Therefore, the jumper is approximately 62.2 meters away from the base of the cliff when she opens her parachute.

Key Concepts

Free Fall

In physics, free fall is a fascinating concept. It occurs when an object moves under the influence of gravitational force alone. In the scenario involving the jumper from El Capitan, she is in a state of free fall right after jumping horizontally off the cliff until she reaches a height of 150 meters above the ground.

Here is how free fall is characterized:

• The only force acting on the object is gravity. This means that other forces, like air resistance, are negligible in this scenario, allowing us to focus solely on gravitational effects.
• Objects in free fall near Earth's surface experience an acceleration of approximately 9.81 meters per second squared, directed downwards.
• During free fall, objects increase their velocity as they descend, because gravity constantly accelerates them downward.

For the jumper, the free fall duration can be determined using the difference in height from the jump-off point to where she opens her parachute. By solving the kinematic equation for vertical motion, we calculate the time of her free fall portion, which is crucial in planning and executing such jumps in extreme sports.

Kinematic Equations

Kinematic equations are mathematical formulas that describe the motion of objects in one, two, or three dimensions. They are particularly useful under constant acceleration conditions, like gravity. These equations allow us to predict various parameters of motion such as time, displacement, velocity, and acceleration.

In the case of free fall off El Capitan, the relevant equation involves the vertical motion:

• \[ h = \frac{1}{2} g t^2 \]

equation, where:

• \( h \) is the vertical displacement (760 meters here as calculated from the exercise).
• \( g \) is the gravitational acceleration (9.81 m/s²).
• \( t \) is the time the jumper is in free fall, which must be solved.

To find how long the jumper is in free fall, solve for \( t \), verifying that the units are consistent to maintain accuracy in calculations. By understanding and using these equations, it becomes easier to predict motion outcomes before they happen, which is immensely valuable for extreme sports planning.

Horizontal Distance

In projectile motion, the horizontal distance, often termed as the "range," is a critical factor to consider. For the jumper running off the cliff, horizontal distance is determined by her horizontal speed and the time she spends in free fall.

Key points about horizontal distance include:

• Horizontal motion is constant and unaffected by gravity since gravity acts vertically.
• The horizontal speed remains constant as there is no acceleration acting sideways (ignoring air resistance).
• The distance can be calculated using the simple formula \[ d = v_{horizontal} \times t \],where \( v_{horizontal} \) is the initial horizontal speed.

In this exercise, the horizontal speed is given as 5.0 m/s, and the jumper is in free fall for approximately 12.44 seconds. These values allow the calculation of the horizontal distance to be approximately 62.2 meters from her jumping point, providing vital information for safety while planning the jump.