A dart gun is aimed horizontally at the center of a large target 150 m away. The initial velocity of the dart is 450 m/s. (Ignore air resistance)

a) Where does the dart strike the target (relative to the center of the target)?

b) To hit the center of the target the dart must be aimed at a point above the center of the target. Find the angle of elevation needed to hit the target.

a. d = Xo*Tf = 150 m.

450 * Tf = 150
Tf = 0.333 s.

d = 0.5g*Tf^2 = 4.9*0.333^2 = 0.544 m.
Below the center.

a) Since the dart gun is aimed horizontally, the initial vertical velocity of the dart is zero. Therefore, we can use the kinematic equation to determine where the dart will strike the target horizontally.

The equation we can use is:

x = v*t

Where:
x = horizontal distance traveled (relative to the center of the target)
v = initial horizontal velocity of the dart (450 m/s)
t = time

To determine the time it takes for the dart to reach the target, we can use the equation:

t = x / v

In this case, the horizontal distance traveled (x) is 150 m and the initial horizontal velocity (v) is 450 m/s. Substituting these values into the equation:

t = 150 m / 450 m/s
t = 1/3 s

Now, we know the time it takes for the dart to hit the target. To determine where it strikes the target horizontally, we can use the equation:

x = v*t

Substituting the values:

x = 450 m/s * (1/3 s)
x = 150 m

Therefore, the dart will strike the target 150 m to the right of the center.

b) To hit the center of the target, the dart must be aimed at a point above the center to compensate for the time it takes to reach the target.

To find the angle of elevation needed, we can use the equation:

y = v0 * t + (1/2) * g * t^2

Where:
y = vertical distance traveled (above the center of the target)
v0 = initial vertical velocity of the dart (unknown)
t = time taken to reach the target (1/3 s)
g = acceleration due to gravity (-9.8 m/s^2)

The vertical distance traveled (y) should be zero in order to hit the center of the target. Therefore, the equation can be rewritten as:

0 = v0 * (1/3 s) + (1/2) * (-9.8 m/s^2) * (1/3 s)^2

Simplifying the equation:

0 = v0/3 - 1/6

To solve for v0, we can multiply both sides of the equation by 3:

0 = v0 - 1/2

Therefore, v0 = 1/2 m/s.

To find the angle of elevation, we can use the trigonometric relationship:

tan(theta) = v0 / v

Where:
theta = angle of elevation
v0 = initial vertical velocity of the dart (1/2 m/s)
v = initial horizontal velocity of the dart (450 m/s)

Substituting the values:

tan(theta) = (1/2 m/s) / (450 m/s)

Taking the inverse tan (arctan) of both sides to find theta:

theta = arctan((1/2 m/s) / (450 m/s))

Using a calculator, we find that theta is approximately 0.062 degrees.

Therefore, to hit the center of the target, the dart must be aimed at an angle of approximately 0.062 degrees above the center.

To answer both parts of the question, we can use the equations of motion for projectile motion. Let's go step by step:

Step 1: Analyzing the horizontal motion
Since the dart is being aimed horizontally, its initial velocity (V₀x) in the horizontal direction will be equal to the dart's actual velocity of 450 m/s. Since there is no horizontal acceleration, the dart will maintain this velocity throughout its flight.

Step 2: Analyzing the vertical motion
The dart will experience vertical acceleration due to gravity (g = 9.8 m/s²). The initial vertical velocity (V₀y) is 0, and we need to find the time it takes for the dart to strike the target.

Step 3: Finding the time of flight
We can use the formula for the time of flight (t):
t = 2 * V₀y / g
Since V₀y is 0, the time of flight will also be 0.

Step 4: Finding the horizontal distance traveled
The horizontal distance traveled (d) can be calculated using the formula:
d = V₀x * t
Substituting the known values:
d = 450 m/s * 0 s = 0 m

a) Answer:
The dart will strike the target at a horizontal distance of 0 m from the center of the target. This means it will hit the center of the target.

b) To find the angle of elevation needed to hit the target, we can use the formula for the angle of launch (θ):
θ = atan(V₀y / V₀x)
Since V₀y is 0, the angle of elevation will also be 0 degrees.

b) Answer:
To hit the center of the target, no angle of elevation is required when the dart is aimed horizontally.