A stone initially at rest is dropped from a 72.4 m tall cliff and travels in free fall. What is the final velocity of the stone just before it hits the ground?
[tex]v_{f}=[?] m / s[/tex]
Do not account for air resistance. Remember, velocity downward is a negative vector (-).
Answer (2)
We analyze the problem and identify the given values: initial velocity ( v i = 0 m/s), height ( h = 72.4 m), and acceleration due to gravity ( g = 9.8 m/s 2 ).
We use the kinematic equation v f 2 = v i 2 + 2 g h to find the final velocity.
Substituting the values, we get v f 2 = 0 2 + 2 × 9.8 × 72.4 = 1419.04 .
Taking the square root and considering the downward direction, the final velocity is − 37.67 s m .
Explanation
Understanding the Problem We are given a stone dropped from a cliff of height 72.4 m. We need to find the final velocity of the stone just before it hits the ground, neglecting air resistance. The stone starts from rest, so its initial velocity is 0 m/s. We know the acceleration due to gravity is approximately 9.8 m/s 2 . Since the stone is falling downwards, we consider the final velocity to be negative.
Choosing the Right Equation We can use the following kinematic equation to find the final velocity: v f 2 = v i 2 + 2 g h where: v f is the final velocity, v i is the initial velocity, g is the acceleration due to gravity (9.8 m/s 2 ), h is the height of the cliff (72.4 m).
Solving for Final Velocity Substitute the given values into the equation: v f 2 = 0 2 + 2 ( 9.8 s 2 m ) ( 72.4 m ) v f 2 = 0 + 1419.04 s 2 m 2 v f = ± 1419.04 s 2 m 2 Since the stone is moving downwards, we take the negative root: v f = − 1419.04 s m v f ≈ − 37.67 s m
Final Answer The final velocity of the stone just before it hits the ground is approximately -37.67 m/s. The negative sign indicates that the velocity is directed downwards.
Examples
Imagine you're designing a safety net for construction workers. Knowing the final velocity of a falling object (like a tool or a piece of equipment) from a certain height is crucial to ensure the net can withstand the impact. By calculating the final velocity using the principles of free fall, engineers can choose materials and designs that provide adequate protection, preventing injuries. This calculation helps in designing effective safety measures in various scenarios, from construction sites to amusement park rides.
The final velocity of a stone dropped from a height of 72.4 m, just before it hits the ground, is approximately -37.67 m/s. This negative value indicates the downward direction of the velocity. The calculation uses the kinematic equation for free fall, considering the acceleration due to gravity.
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