Crickets Chirpy and Milada jump from the top of a vertical cliff. Chirpy drops downward and reaches the ground in 2.70 s, while Milada jumps horizontally with an initial speed of 95.0 cm/s. How far from the base of the cliff will Milada hit the ground? Ignore air resistance.

Verified step by step guidance

1. First, understand that Chirpy and Milada are both subject to gravitational acceleration, which is approximately 9.81 m/s² downward. Chirpy drops vertically, while Milada jumps horizontally.
2. Calculate the height of the cliff using Chirpy's fall time. Since Chirpy falls vertically, use the formula for free fall: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>h</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>g</mi><msup><mi>t</mi><mn>2</mn></msup></mrow></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is the acceleration due to gravity and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time.
3. Convert Milada's initial horizontal speed from cm/s to m/s for consistency in units. Since 1 m = 100 cm, Milada's speed is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>95.0</mn><mn>100</mn></mfrac></math> m/s.
4. Determine the time it takes for Milada to reach the ground. Since both crickets fall from the same height, Milada will also take 2.70 s to reach the ground.
5. Calculate the horizontal distance Milada travels using the formula: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>d</mi><mo>=</mo><mi>v</mi><mi>t</mi></mrow></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> is Milada's horizontal speed and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time.

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. In this scenario, Milada's horizontal jump is a classic example of projectile motion, where the horizontal and vertical motions are independent of each other. The horizontal distance traveled can be calculated using the initial horizontal velocity and the time of flight.

Free Fall

Free fall describes the motion of an object under the influence of gravitational force only. Chirpy's drop is a free fall, taking 2.70 seconds to reach the ground. This time is crucial for determining how long Milada is in the air, as both crickets experience the same vertical motion due to gravity, allowing us to calculate the horizontal distance Milada travels.

Independence of Motion

The independence of motion principle states that horizontal and vertical motions are independent of each other. In this problem, Milada's horizontal motion does not affect her vertical fall. Thus, the horizontal distance she covers is determined by her initial horizontal speed and the time it takes to fall, which is the same as Chirpy's fall time, 2.70 seconds.