For its size, the common flea is one of the most accomplished jumpers in the animal world. A 2.1 mm-long, 0.46 mg critter can reach a height of 17 cm in a single leap.
A) Neglecting air drag, what is the takeoff speed of such a flea?
I don't understand how I can find the velocity without being given the elapsed time, and I'd appreciate any help. Thank you.
Well, you will have to assume that the flea goes straight up.
The easy way to do it is to say the kinetic energy at takeoff equals the potential energy at the top when the speed is zero.
(1/2) m v^2 = m g h
or
v^2 = 2 g h
here g = 9.8 m/s^2
h = .17 meters
To find the takeoff speed of the flea, you're correct that you need to know the elapsed time. However, in this particular question, we can use the information provided to calculate the takeoff speed without explicitly knowing the time it takes for the flea to jump.
Here's how you can approach this problem:
1. Start by assuming that the flea's vertical motion is solely determined by the force of gravity acting on it after takeoff.
2. Use the equation of motion for vertical motion under constant acceleration:
s = ut + (1/2)at^2
where:
- s is the displacement (change in height),
- u is the initial velocity (takeoff speed),
- t is the time, and
- a is the acceleration (due to gravity).
3. Rearrange the equation to solve for takeoff speed (u):
u = (2s / t)^(1/2)
In this case, the flea reaches a height of 17 cm (or 0.17 m). However, we don't know the time it takes for the flea to reach this height. To overcome this, we can consider the fact that the time it takes for an object to reach its maximum height is equal to the time it takes for that object to fall back down to its original height.
Since the flea jumps straight up and then falls back down to its original position, we can consider its total displacement to be zero.
0 = 0.17 m + (-0.17 m) (upwards displacement + downwards displacement)
Now, plug in the values into the equation for takeoff speed:
u = (2s / t)^(1/2)
u = (2 * 0.17 m / t)^(1/2)
Since s = 0 and we don't know the time (t), we can't solve for the takeoff speed (u) without more information.
Therefore, in this specific case, without knowing the elapsed time or any additional information, it is not possible to calculate the takeoff speed of the flea.