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.
Find a time that the dart moves inside the barrel.
Express your answer in terms of the variables L and v.
If the dart's exit speed is 16.0 m/s and the length of the blowgun is 1.10 m , find the time the dart is in the barrel.
Express your answer with the appropriate units.
Why did the dart go to a comedy show? Because it wanted to improve its barrel performance!
To find the time the dart is in the barrel, we can use the formula:
time = distance / speed
In this case, the distance is the length of the blowgun barrel (L) and the speed is the exit speed of the dart (v). So, the time is:
time = L / v
Plugging in the values given, we have:
time = 1.10 m / 16.0 m/s = 0.06875 s
So, the dart is in the barrel for approximately 0.06875 seconds.
To find the time the dart is in the barrel, we can use the equation of motion:
v = u + at
where
v is the final velocity (exit speed),
u is the initial velocity (which is 0 for the dart inside the barrel),
a is the acceleration (uniform),
and t is the time.
Since the initial velocity is 0 and the final velocity is given as 16.0 m/s, we can write:
16.0 m/s = 0 + a * t
To find the acceleration, we can use the equation:
a = Δv / Δt
where
Δv is the change in velocity (16.0 m/s - 0 = 16.0 m/s),
and Δt is the change in time (the time the dart is in the barrel).
Substituting the values into the equation, we have:
16.0 m/s = (16.0 m/s) / Δt
Simplifying, we can cancel out the meters per second:
1 = 1 / Δt
Multiplying both sides by Δt, we get:
Δt = 1 s
Therefore, the time the dart is in the barrel is 1 second.
To find the time the dart is in the barrel, we can use the equation of motion:
v = u + at
where:
v = final velocity of the dart (exit speed) = 16.0 m/s
u = initial velocity of the dart (inside the barrel) = 0 m/s (dart starts from rest)
a = acceleration of the dart = ? (to be determined)
t = time the dart is in the barrel = ? (to be determined)
Since the acceleration is uniform, we can use the equation:
a = (v - u) / t
Substituting the given values:
a = (16.0 m/s - 0 m/s) / t
Simplifying:
a = 16.0 m/s / t
Now, we need to find the acceleration of the dart in the barrel. Since the acceleration is uniform, it can be calculated using the formula:
a = Δv / Δt
where:
Δv = change in velocity = final velocity - initial velocity = 16.0 m/s - 0 m/s = 16.0 m/s
Δt = change in time = time to travel the length of the barrel = ? (to be determined)
Since the dart starts from rest inside the barrel, the initial velocity is 0 m/s. The final velocity is given as 16.0 m/s.
We can substitute these values into the formula for acceleration:
a = Δv / Δt
= (16.0 m/s - 0 m/s) / Δt
= 16.0 m/s / Δt
Now we can equate this expression for acceleration with the previous expression:
16.0 m/s / t = 16.0 m/s / Δt
Since both sides of the equation are equal to the same value of acceleration, we can set them equal to each other:
16.0 m/s / t = 16.0 m/s / Δt
To solve for Δt, we can cross multiply:
(16.0 m/s)(t) = (16.0 m/s)(Δt)
Dividing both sides by 16.0 m/s:
t = Δt
Therefore, the time the dart is in the barrel, t, is equal to the time it takes for the dart to travel the length of the barrel, Δt.
The length of the blowgun barrel, L, is given as 1.10 m. So, Δt = L / v.
Substituting the values:
Δt = 1.10 m / 16.0 m/s
Simplifying:
Δt = 0.06875 s
Therefore, the time the dart is in the barrel, t, is approximately 0.06875 seconds.
It started at v = 0
It exited at v = v
v versus t is LINEAR (constant acceleration)
Therefore Vaverage = v/2
L = Vaverage * t
so
t = L/Vaverage = 2 L / v
t = 2 * 1.10 / 16.0