Sure! Here's a classic projectile motion problem involving an object falling into a body of water:
Problem: A stone is dropped from the top of a cliff that is 78.4 meters high. Assuming there is no air resistance, calculate the time it takes for the stone to hit the water (the body of water at the base of the cliff). Also, compute the speed of the stone just before it hits the water.
Solution:
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Calculate the time of fall: We can use the following kinematic equation: \[ d = v_i t + \frac{1}{2} a t^2 \] where:
- \(d\) is the distance fallen (78.4 m),
- \(v_i\) is the initial velocity (0 m/s, since the stone is dropped),
- \(a\) is the acceleration due to gravity (approximately \(9.81 , \text{m/s}^2\)),
- \(t\) is the time in seconds.
Plugging in the values, we have: \[ 78.4 = 0 \cdot t + \frac{1}{2} (9.81) t^2 \] This simplifies to: \[ 78.4 = 4.905 t^2 \] Solving for \(t^2\): \[ t^2 = \frac{78.4}{4.905} \approx 16 \] Taking the square root to find \(t\): \[ t \approx 4 , \text{s} \]
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Calculate the speed just before hitting the water: We can use the following kinematic equation to find the final velocity (\(v_f\)): \[ v_f = v_i + a t \] Again, substituting in the values: \[ v_f = 0 + (9.81)(4) \approx 39.24 , \text{m/s} \]
So, the stone takes approximately 4 seconds to hit the water, and its speed just before hitting the water is approximately 39.24 m/s.
This scenario exemplifies the principles of projectile motion, particularly the free fall and the effect of gravity on an object in motion.