For its size, the common flea is one of the most accomplished jumpers in the animal world. A 2.30 mm -long, 0.540 mg critter can reach a height of 18.0 cm in a single leap.
a) Neglecting air drag, what is the takeoff speed of such a flea?
-i got 1.88m/s
b) Calculate the kinetic energy of this flea at takeoff and its kinetic energy per kilogram of mass.
-wouldn't we just use KE = 1/2mv^2 cause i got 9.54e-4 J but it says my answer is wrong
nvm i got it, kept changing mg to g instead of kg
a. V^2 = Vo^2 + 2g*h = 0.
Vo^2 = -2g*h = -(-19.6)*0.18 = 3.53, Vo = 1.88 m/s.
b. KE = 0.5M*Vo^2 = 0.5*0.54*10^-6*1.88^2 = 9.54*10^-7 Joules.
To correctly calculate the kinetic energy (KE) of the flea, we need to convert the mass from milligrams to kilograms.
a) Neglecting air drag, the takeoff speed (v) of the flea can be calculated using the equation for vertical motion:
v^2 = u^2 + 2as
where u is the initial velocity (takeoff speed), a is acceleration (gravity, approximately 9.8 m/s^2), and s is the displacement (18.0 cm or 0.18 m).
Rearranging the equation, we have:
u = √(v^2 - 2as)
Plugging in the given values:
u = √(0^2 - 2 * 9.8 * 0.18)
u = √(-35.56)
u ≈ 1.88 m/s
Therefore, the takeoff speed of the flea is approximately 1.88 m/s.
b) To calculate the kinetic energy (KE) of the flea at takeoff, we need to use the formula:
KE = (1/2) * m * v^2
where m is the mass of the flea (0.540 mg or 0.00054 kg) and v is the takeoff speed (1.88 m/s).
Plugging in the given values:
KE = (1/2) * 0.00054 * (1.88)^2
KE ≈ 9.59 × 10^(-4) J
The kinetic energy of the flea at takeoff is approximately 9.59 × 10^(-4) J.
To find the kinetic energy per kilogram of mass, simply divide the kinetic energy by the mass of the flea:
Kinetic Energy per kilogram = KE / m
KE/kg = 9.59 × 10^(-4) / 0.00054
KE/kg ≈ 1.77 J/kg
Therefore, the kinetic energy per kilogram of mass of the flea is approximately 1.77 J/kg.
To solve this problem, we'll need to use the formulas for kinetic energy (KE) and gravitational potential energy (PE) to determine the takeoff speed and the kinetic energy of the flea.
a) To find the takeoff speed of the flea, we can use the equation for gravitational potential energy:
PE = mgh
Where PE is the potential energy, m is the mass of the flea, g is the acceleration due to gravity (9.8 m/s^2), and h is the height reached in the jump.
Rearranging the equation, we get:
h = PE / (mg)
Plugging in the given values:
h = 0.18 m
m = 0.540 mg = 0.00054 g
g = 9.8 m/s^2
We can now calculate the potential energy:
PE = (0.00054 g) * (9.8 m/s^2) * (0.18 m)
PE = 0.00094 J
Now, we can find the takeoff speed using the formula for kinetic energy:
KE = 1/2 * m * v^2
Rearranging the equation, we get:
v = sqrt((2 * KE) / m)
Plugging in the given values for KE and m, we obtain:
v = sqrt((2 * 0.00094 J) / (0.00054 g))
v = 1.85 m/s
Therefore, the takeoff speed of the flea is approximately 1.85 m/s, which is different from your answer of 1.88 m/s.
b) To calculate the kinetic energy of the flea at takeoff, we can again use the formula for kinetic energy:
KE = 1/2 * m * v^2
Plugging in the given values for m and v, we get:
KE = 1/2 * (0.00054 g) * (1.85 m/s)^2
KE = 9.54e-4 J
So, your calculated kinetic energy of 9.54e-4 J is correct.
However, to find the kinetic energy per kilogram of mass, we need to divide the kinetic energy by the mass of the flea. In this case, we need to convert the mass to kilograms.
m = 0.540 mg = 0.540 × 10^(-6) kg
Now we can calculate the kinetic energy per kilogram:
KE per kg = KE / m
KE per kg = (9.54e-4 J) / (0.540 × 10^(-6) kg)
KE per kg = 1.76 J/kg
So, the kinetic energy per kilogram of mass for the flea at takeoff is 1.76 J/kg.