A dart leaves the barrel of a blowgun at a speed v. The length of the blowgun barrel is L. Assume that the acceleration of the dart in the barrel is uniform. If the dart's exit speed is 15.0 m/s and the length of the blowgun is 1.50 m , find the time the dart is in the barrel.
To find the time the dart is in the barrel, we can use the equation of motion:
v = u + at
Where:
v = final velocity (exit speed) = 15.0 m/s
u = initial velocity (0 m/s since the dart starts from rest)
a = acceleration (assumed constant)
t = time in seconds
We know that the dart accelerates uniformly in the barrel, so the acceleration (a) is constant. We can find the acceleration using the formula:
v^2 = u^2 + 2as
Rearranging the formula to solve for acceleration (a):
a = (v^2 - u^2) / 2s
a = (15.0^2 - 0) / (2 * 1.50)
a = 112.5 / 3
a = 37.5 m/s^2
Now that we have the acceleration, we can find the time the dart is in the barrel using the first equation:
15.0 = 0 + 37.5 * t
t = 15.0 / 37.5
t = 0.4 seconds
Therefore, the dart is in the barrel for 0.4 seconds before exiting at a speed of 15.0 m/s.