Fleas can jump quite high; experiments show that they reach an altitude of 130 times their own height when they jump vertically! Suppose instead that a flea can jump so that it lands a maximum distance of 19.60 cm away. What is the take-off speed for this flea?
Well, it sounds like this flea is quite the high-jumper! In order to find the take-off speed, we need to consider the relationship between the horizontal distance and the vertical height of the flea's jump.
Now, since the flea jumps a maximum distance of 19.60 cm horizontally, we can use this information to calculate how high it jumps. We know that the ratio between the horizontal and vertical distances is equal to 130.
So, if we let "h" represent the height of the flea's jump, we can set up the equation:
19.60 cm / h = 130
To find h, we can solve this equation for h:
h = 19.60 cm / 130
h ≈ 0.15 cm
Now, we know that the take-off speed of the flea can be determined by the formula:
v = √(2gh)
Where "g" represents the acceleration due to gravity (approximately 9.8 m/s²).
Since we've been working with centimeters throughout, let's convert h to meters:
h ≈ 0.15 cm = 0.0015 m
Now we can plug the values into the formula:
v = √(2 * 9.8 m/s² * 0.0015 m)
v ≈ √(0.0294) ≈ 0.171 m/s
So, the take-off speed for this impressive jumping flea is approximately 0.171 meters per second. That flea sure knows how to "spring" into action!
To find the take-off speed of the flea, we can use the principle of conservation of energy.
When the flea takes off, it converts its potential energy into kinetic energy. At the highest point of its trajectory, when it is about to reach maximum distance, it has no potential energy left and only kinetic energy.
The potential energy (PE) of an object at a certain height is given by the formula PE = m * g * h, where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
The kinetic energy (KE) of an object is given by the formula KE = 0.5 * m * v², where v is the velocity of the object.
Since the flea reaches maximum distance when it has no potential energy left, we can equate the potential energy at take-off to the kinetic energy at maximum distance:
m * g * h = 0.5 * m * v²
The mass of the flea cancels out, so we're left with:
g * h = 0.5 * v²
Now let's solve the equation for v, the take-off velocity.
v² = 2 * g * h
Substituting the known values, with g = 9.8 m/s² and h = 19.60 cm = 0.196 m:
v² = 2 * 9.8 m/s² * 0.196 m
v² = 3.84 m²/s²
Taking the square root of both sides, we find:
v ≈ 1.96 m/s
Therefore, the take-off speed for this flea is approximately 1.96 m/s.