You buy a plastic dart gun, and being a clever physics student you decide to do a quick calculation to find its maximum horizontal range. You shoot the gun straight up, and it takes 3.8 s for the dart to land back at the barrel. What is the maximum horizontal range of your gun?
it wil go ro the ski
hf=hi+vi*t-1/2 g t^2
if hf=hi=0
0=t(vi-1/2 g t)
so t= 2Vi/g
solve for Vi.
Now, max range at 45 deg.
find time in air:
hf=hi+visin45*t-1/2 g t^2 again,hf=hi=0
t= sqrt 2visin45/g
max range: distance= vicos45*t
42.13
16.28
Hi! Sorry the math for the proposed answer is wrong. The equation was manipulated incorrectly and will give a wrong answer
To find the maximum horizontal range of the dart gun, we need to consider the motion of the dart.
First, let's break down the motion into two components: vertical and horizontal. The vertical motion is influenced by gravity, while the horizontal motion is not. Since the question mentions that the dart is shot straight up and lands back at the barrel, we can assume that there is no wind or any external forces affecting the horizontal motion.
Now, let's analyze the vertical motion. When the dart is shot straight up, it travels upward against the force of gravity until it reaches its highest point, where its vertical velocity becomes zero. From that point, it falls back down to the starting position, taking a total time of 3.8 seconds.
Knowing that the time it takes to reach the highest point is half of the total time, we can determine that it takes 3.8/2 = 1.9 seconds for the dart to reach its highest point.
In the vertical motion, the time it takes to reach the highest point is symmetric to the time it takes to fall back down. So, the total time for the dart to reach its highest point and fall back to the barrel is 1.9 + 1.9 = 3.8 seconds.
During this time, the velocity of the dart changes due to gravity. The vertical velocity at the highest point is zero, and we can use this information to calculate the maximum height (H) that the dart reaches.
Using the equation of motion in the vertical direction, we have:
H = (1/2) * g * t^2
where g is the acceleration due to gravity and t is the time it takes to reach the highest point.
Assuming a standard acceleration due to gravity of 9.8 m/s^2, we can plug in the values:
H = (1/2) * 9.8 * (1.9)^2
H ≈ 8.84 meters
Now that we know the maximum height the dart reaches, we can use the concept of symmetry to determine the maximum horizontal range.
The time it takes for the dart to reach the maximum height is 1.9 seconds, and it will take the same amount of time to fall back down. During this time, the horizontal velocity remains constant, as there is no horizontal force acting on the dart.
Using the formula for horizontal distance:
Range = Horizontal velocity * Time
And since we know the time is 1.9 seconds, we can calculate the horizontal velocity by dividing the horizontal distance by the time:
Range = Horizontal velocity * 1.9
Simplifying the equation, we find:
Range = Horizontal velocity * 3.8
To calculate the horizontal velocity, we need to determine the time it takes for the dart to reach its highest point. Since the motion is symmetric, we can divide the total time in half:
t_h = 3.8 / 2
t_h = 1.9 seconds
Finally, substituting back into the equation, we get:
Range = Horizontal velocity * 1.9
Range = Horizontal velocity * (3.8 / 2)
So, the maximum horizontal range of your dart gun is determined by the horizontal velocity of the dart multiplied by 1.9 seconds. Unfortunately, the horizontal velocity is not provided in the question, so we would need extra information to calculate the exact value.