Problem 11 - University Physics with Modern Physics

Question

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.

Step-by-Step Solution

Answer

Milada lands 2.565 meters from the base of the cliff.

Step 1: Recognize the problem type

The problem involves projectile motion for Milada, who jumps horizontally off a cliff. We need to calculate how far from the base of the cliff Milada lands.

Step 2: Understand Milada's vertical motion

Both crickets fall for 2.70 seconds since they start from the same height. This time will be used to calculate the horizontal distance Milada travels.

Step 3: Set up horizontal motion equation

For horizontal motion, use the formula:\[x = v_0 imes t\]where \(x\) is the horizontal distance, \(v_0\) is the initial horizontal speed (95.0 cm/s, convert to meters: 0.95 m/s), and \(t\) is the time (2.70 s).

Step 4: Calculate horizontal distance

Substitute the values into the horizontal motion equation:\[x = 0.95 \, \text{m/s} \times 2.70 \, \text{s} = 2.565 \, \text{m}\]Thus, Milada lands 2.565 meters from the base of the cliff.

Key Concepts

Horizontal Motion Equation

In projectile motion, horizontal motion is straightforward. The key is to remember that horizontal velocity remains constant if air resistance is ignored. This is applicable when an object is projected horizontally from a height, like Milada the cricket. To find how far Milada lands from the cliff, we utilize the horizontal motion equation:

• \( x = v_0 \times t \)

Here, \( x \) is the horizontal distance traveled, \( v_0 \) is the initial horizontal speed, and \( t \) is the time of flight.
Since there are no horizontal forces acting on Milada, the velocity and hence the initial speed (\( v_0 \)) stays the same throughout the flight.
This equation is essential for analyzing horizontal motion in projectile problems and, when applied correctly, allows us to calculate how far an object will land from its starting point on a horizontal path.

Initial Speed Conversion

Before using the horizontal motion equation, it's important to ensure unit consistency.
Milada's initial speed is given as 95.0 cm/s, which needs to be converted to meters per second for appropriate unit alignment in the equation:

• Conversion factor: 1 cm = 0.01 m
• Therefore, 95.0 cm/s × 0.01 = 0.95 m/s

This conversion ensures that every parameter in the formula \( x = v_0 \times t \) uses the metric system, providing an accurate computation of the horizontal distance.

Time of Flight

Time of flight is crucial in determining how far a projectile travels horizontally.
In this problem, both Chirpy and Milada experience the same time of flight – 2.70 seconds – as they fall from the cliff. This is because they both start their motion from the same height and are subject to gravity.
Understanding the time of flight involves analyzing the vertical motion aspect of projectile motion, which is influenced only by gravitational acceleration when air resistance is negligible.

• Gravity doesn't affect horizontal motion directly.
• Thus, once the time of flight is known from vertical analysis, it can be applied in the horizontal motion equation:
• Utilize \( x = v_0 \times t \) with \( t = 2.70 \) seconds

This knowledge lets us calculate that Milada impacts the ground 2.565 meters from the base of the cliff, primarily dictated by the time she spends in the air.