An egg initially at rest is dropped from the roof of a 31.5 m tall building and travels in free fall.
What is the final velocity of the egg just before it hits the ground?
[tex]v_{f}=[?] m / s[/tex]
Do not account for air resistance.
Remember, downward velocity is a negative vector (-).
Answer (2)
We use the kinematic equation v f 2 ā = v i 2 ā + 2 g h .
Substitute the given values: v i ā = 0 , g = 9.8 m/s 2 , and h = 31.5 m.
Calculate v f 2 ā = 2 ( 9.8 ) ( 31.5 ) = 617.4 .
Solve for v f ā : v f ā = ā 617.4 ā \[ 0.5 e x ] v f ā = ā 24.84753508901839 . The final velocity is ā 24.85 ā m/s.
Explanation
Problem Analysis We are given that an egg is dropped from a height of 31.5 meters. We need to find the final velocity of the egg just before it hits the ground, neglecting air resistance. Since the egg is dropped, its initial velocity is 0 m/s. The acceleration due to gravity is 9.8 m/s 2 . We will use a kinematic equation to solve for the final velocity.
Kinematic Equation We will use the following kinematic equation: v f 2 ā = v i 2 ā + 2 g h where: v f ā is the final velocity, v i ā is the initial velocity, g is the acceleration due to gravity (9.8 m/s 2 ), h is the height (31.5 m).
Substitution Since the egg is dropped from rest, v i ā = 0 m/s. Substituting the given values into the equation, we get: v f 2 ā = 0 2 + 2 ( 9.8 ) ( 31.5 ) v f 2 ā = 2 ( 9.8 ) ( 31.5 ) v f 2 ā = 617.4
Solving for Final Velocity Now, we solve for v f ā by taking the square root of both sides. Since the egg is moving downwards, we take the negative square root: v f ā = ā 617.4 ā v f ā = ā 24.84753508901839 The final velocity of the egg just before it hits the ground is approximately -24.85 m/s.
Final Answer Therefore, the final velocity of the egg just before it hits the ground is approximately -24.85 m/s.
Examples
Understanding the final velocity of a falling object is crucial in many real-world scenarios, such as engineering design and safety analysis. For instance, when designing packaging for fragile items, engineers need to calculate the impact velocity to ensure the packaging can withstand the force. Similarly, in construction, knowing the potential impact velocity of falling debris helps in designing safety measures to protect workers and the public. This principle is also fundamental in sports science, where analyzing the velocity of a ball or an athlete helps optimize performance and prevent injuries.
The final velocity of the egg just before it hits the ground, when dropped from a height of 31.5 meters, is approximately -24.85 m/s. This value is negative because the egg is in free fall, moving downward towards the ground. We calculate this using the kinematic equation for uniformly accelerated motion, accounting for an initial velocity of 0 m/s and an acceleration due to gravity of 9.8 m/sĀ².
;
