A typical flea jumps vertically w/an initial velocity of about 5ft/s. How much does the flea jump?
Ans. 65/64ft or 4.7in.. pls show the solution and explain.. thanks a lot.. :)
If H is the height jumped and V is the initial velocity (vertical), conservation of energy tells you that
M g H = (1/2) M V^2
Mass (M) cancels, giving you
H = V^2/(2g)
g = 32.2 ft/s^2, so
H = 25/(64.4) = 0.388 ft = 4.66 inches
Your stated answer of 65/64 should be 25/64 ft. They are expecting you to use 32 ft/s^2 for g, which is not quite right.
To find the distance the flea jumps, we can use basic kinematics equations. The equation that relates distance, initial velocity, time, and acceleration for an object in free fall is:
d = (1/2) * a * t^2 + V₀ * t
Where:
d is the distance traveled,
a is the acceleration (which for free fall is equal to gravity: -32.174 ft/s^2),
t is the time elapsed, and
V₀ is the initial velocity.
In this case, the flea jumps vertically, so the time it takes to reach the maximum height will be the same as the time it takes to fall back down. Therefore, we can calculate the total distance traveled by the flea by multiplying the distance traveled during the ascent (upward jump) by 2.
For the upward jump:
V₀ = 5 ft/s
a = -32.174 ft/s^2 (acceleration due to gravity)
t = ? (unknown)
Now, let's find the time it takes for the flea to reach the maximum height. We can use the kinematic equation that relates final velocity, initial velocity, acceleration, and time:
V = V₀ + a * t
At the maximum height, the flea temporarily stops moving (V = 0), so the equation becomes:
0 = 5 ft/s + (-32.174 ft/s^2) * t
Solving the equation for t:
-5 ft/s = -32.174 ft/s^2 * t
t = -5 ft/s / (-32.174 ft/s^2)
t ≈ 0.1554 s
Now that we know the time for the upward jump, we can substitute it back into the original equation to find the distance traveled during the ascent:
d₁ = (1/2) * (-32.174 ft/s^2) * (0.1554 s)^2 + 5 ft/s * 0.1554 s
d₁ ≈ -1.036 ft + 0.777 ft
d₁ ≈ -0.259 ft
The negative sign indicates that the displacement during the ascent is in the opposite direction of gravity, which makes sense for upward motion.
To find the total distance traveled by the flea, we multiply the displacement during the ascent by 2:
d_total = 2 * -0.259 ft
d_total ≈ -0.518 ft
Again, the negative sign signifies the upward direction of the jump.
Converting the distance to a positive value:
d_total ≈ 0.518 ft
Lastly, converting the distance to inches:
0.518 ft * 12 in/1 ft ≈ 6.216 in
Therefore, a typical flea jumps about 6.216 inches or approximately 4.7 inches.