If a flea can jump straight up to a height of 0.310 m, what is its initial speed as it leaves the ground?
How long is it in the air?
To find the initial speed of the flea, we can use the principles of projectile motion. The height a projectile reaches is influenced by the initial speed and the acceleration due to gravity.
The formula to calculate the height reached by a projectile is given by:
h = (v_0^2 * sin^2(θ)) / (2 * g)
Where:
h = height reached (0.310 m in this case)
v_0 = initial speed of the flea (what we want to find)
θ = launch angle (for a vertical jump, θ is 90 degrees)
g = acceleration due to gravity (approximately 9.8 m/s^2)
Substituting the known values into the equation:
0.310 = (v_0^2 * sin^2(90)) / (2 * 9.8)
Since sin(90) is equal to 1, the equation simplifies to:
0.310 = (v_0^2) / (2 * 9.8)
Now we can solve for v_0:
v_0^2 = 0.310 * 2 * 9.8
v_0^2 = 6.076
v_0 = √6.076
v_0 ≈ 2.47 m/s
Therefore, the initial speed of the flea as it leaves the ground is approximately 2.47 m/s.
To find how long the flea is in the air, we can use the equation for the time of flight of a projectile:
t = (2 * v_0 * sin(θ)) / g
Since the flea jumps straight up, the launch angle (θ) is 90 degrees. Substituting the known values:
t = (2 * 2.47 * sin(90)) / 9.8
t = (4.94) / 9.8
t ≈ 0.50 seconds
Therefore, the flea is in the air for approximately 0.50 seconds.