An egg is dropped from a height of 90 ft. If acceleration due to gravity is [tex]$-16 ft / s ^2$[/tex] and initial velocity is [tex]$0 ft / s$[/tex], approximately how long will it take for the egg to reach the ground?
[tex]$h(t)=a t^2+v t+h_0$[/tex]
A. 1.9 s
B. 2.4 s
C. 9.7 s
D. 40.7 s
Answer (2)
Set up the height function: h ( t ) = − 16 t 2 + 90 .
Set h ( t ) = 0 to find the time when the egg reaches the ground: − 16 t 2 + 90 = 0 .
Solve for t : t = 16 90 .
Calculate the approximate value of t : t ≈ 2.4 s .
Explanation
Understanding the Problem We are given the height function h ( t ) = a t 2 + v t + h 0 , where:
a = − 16 f t / s 2 is the acceleration due to gravity,
v = 0 f t / s is the initial velocity,
h 0 = 90 f t is the initial height. We want to find the time t when the egg reaches the ground, i.e., when h ( t ) = 0 .
Setting up the Equation We need to solve the equation h ( t ) = 0 for t . Plugging in the given values, we have: − 16 t 2 + 0 t + 90 = 0
Simplifying the Equation Simplifying the equation, we get: − 16 t 2 + 90 = 0
Isolating t 2 Now, we solve for t 2 :
16 t 2 = 90
Solving for t 2 Divide both sides by 16: t 2 = 16 90
Solving for t Take the square root of both sides to solve for t :
t = 16 90 Since time cannot be negative, we only consider the positive square root.
Calculating the Time Calculating the value of t :
t = 16 90 ≈ 2.37
Final Answer Therefore, it will take approximately 2.37 seconds for the egg to reach the ground. Among the given options, 2.4 s is the closest approximation.
Examples
Understanding how long it takes for an object to fall is crucial in many real-world scenarios. For example, engineers designing safety equipment need to calculate the impact time of falling objects to ensure the equipment can withstand the force. Similarly, in sports, understanding the trajectory and fall time of a ball helps athletes improve their performance. Knowing the acceleration due to gravity and initial conditions allows us to predict these fall times accurately using physics principles. The formula h ( t ) = a t 2 + v t + h 0 is a fundamental tool in these calculations, where h ( t ) is the height at time t , a is the acceleration, v is the initial velocity, and h 0 is the initial height. This knowledge is also applicable in fields like construction, where predicting the fall time of materials is essential for safety and efficiency.
The egg dropped from a height of 90 feet will take approximately 2.4 seconds to reach the ground, calculated using the height function. The equation h ( t ) = − 16 t 2 + 90 leads to setting h ( t ) = 0 and solving for t . The answer choice is B. 2.4 s.
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