Calculating the Height of a Cliff Using Horizontal Projectile Motion

Imagine a scenario where a ball is thrown horizontally from the top of a cliff with a speed of 24 meters per second. The ball hits the ground 5.0 seconds later. The question is, approximately how high is the cliff? This problem can be solved by utilizing the principles of mechanics, specifically the kinematic equations of motion for free fall.

The vertical motion of the ball is influenced by gravity. To find the height of the cliff, we can use the kinematic equation:

h (frac{1}{2} g t^2)

Where:

n- h is the height in meters n- g is the acceleration due to gravity, approximately 9.81 m/s2 n- t is the time in seconds (in this case, 5.0 seconds)

Substitute the values into the equation:

h  (frac{1}{2} times 9.81 text{ m/s}^2 times 5.0^2 text{ s}^2)

Let's break down the calculation step-by-step:

Calculate (5.0^2 text{ s}^2): 5.02 25.0 s2 Calculate (frac{1}{2} times 9.81 text{ m/s}^2): (frac{1}{2} times 9.81 4.905 text{ m/s}^2) Multiply (4.905 text{ m/s}^2) by 25.0 s2: (h 4.905 times 25.0 122.625 text{ m})

Thus, the height of the cliff is approximately 122.6 meters, considering the calculations and rounding to a practical level.

Additional Considerations and Variations

It's important to note that the horizontal velocity of 24 m/s is not relevant to the height calculation, as it does not affect the vertical component of the motion. The formula for acceleration under standard gravity g is:

y gt2

Where:

n- y is the vertical distance n- g is the acceleration due to gravity (-9.80665 m/s2) n- t is the time in seconds

Using the formula, if we assume a different time or gravity:

Using g -9.80665 m/s2 and t 5.72 s: (y frac{1}{2} times (-9.80665) times 5.72^2 159.30902925 text{ m}) Rounding to 3 significant figures: 159 m or (1.59 times 10^2 text{ m}) Using g 9.8 m/s2 and t 3 s: (s frac{1}{2} times 9.8 times 3^2 44.1 text{ m}) Another scenario with g 9.8 m/s2 and t 7 s: (h frac{1}{2} times 9.8 times 7^2 240.1 text{ m}) Horizontal distance dh 210 m if vh 30 m/s and t 7 s

These variations help in understanding the range of scenarios in which the height of a cliff or object can be calculated using the principles of horizontal projectile motion and free fall equations.

Conclusion

As demonstrated, calculating the height of a cliff using the principles of horizontal projectile motion and kinematic equations is a straightforward process. By understanding and utilizing these principles, one can accurately determine distances and heights in various physical scenarios.