You attach a meter stick to an oak tree, such that the top of the meter stick is 2.47 meters above the ground. Later, an acorn falls from somewhere higher up in the tree. If the acorn takes 0.166 seconds to pass the length of the meter stick, how high above the ground was the acorn before it fell (a…
The correct answer and explanation is:
To determine how high the acorn was above the ground before it fell, we can use kinematic equations for motion under gravity. Here’s the step-by-step solution:
Known values:
- Length of the meter stick: Δy=1.00 m\Delta y = 1.00 \, \text{m}
- Time taken to pass the meter stick: t=0.166 st = 0.166 \, \text{s}
- Acceleration due to gravity: g=9.81 m/s2g = 9.81 \, \text{m/s}^2
- Height of the top of the meter stick from the ground: hstick top=2.47 mh_{\text{stick top}} = 2.47 \, \text{m}
Step 1: Initial velocity of the acorn at the top of the meter stick
The acorn’s motion while passing the meter stick involves both its initial velocity (viv_i) and the constant acceleration due to gravity. Using the kinematic equation: Δy=vit+12gt2\Delta y = v_i t + \frac{1}{2} g t^2
Substitute the known values: 1.00=vi(0.166)+12(9.81)(0.166)21.00 = v_i (0.166) + \frac{1}{2} (9.81) (0.166)^2 1.00=0.166vi+0.1351.00 = 0.166 v_i + 0.135 0.865=0.166vi0.865 = 0.166 v_i vi≈5.21 m/sv_i \approx 5.21 \, \text{m/s}
Step 2: Height of the acorn above the top of the meter stick
The acorn’s velocity at the top of the meter stick comes from falling from a higher point. Using: vi2=2ghv_i^2 = 2 g h
Substitute vi=5.21 m/sv_i = 5.21 \, \text{m/s}: (5.21)2=2(9.81)h(5.21)^2 = 2 (9.81) h 27.14=19.62h27.14 = 19.62 h h≈1.38 mh \approx 1.38 \, \text{m}
Step 3: Total height above the ground
The total height above the ground is: htotal=hstick top+h=2.47+1.38=3.85 mh_{\text{total}} = h_{\text{stick top}} + h = 2.47 + 1.38 = 3.85 \, \text{m}
Final Answer:
The acorn was 3.85 meters above the ground before it fell.
Explanation:
This problem involves free fall motion, where gravity accelerates the object downward. We broke the problem into two parts: finding the velocity of the acorn as it passes the top of the meter stick, and determining the height from which it started falling. By solving these equations step by step, we accounted for both velocity and acceleration due to gravity, ensuring accuracy in determining the total height above the ground. This methodology highlights the interplay of time, distance, and velocity in kinematic problems.