You are practicing throwing darts in your dorm. You stand 3.21 m from the wall on which the board hangs. The dart leaves your hand with a horizontal velocity at a point 1.94 m above the ground. The dart strikes the board at a point 1.73 m from the ground.
a) Calculate the time of flight of the dart.
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b) Calculate the initial speed of the dart.
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c) Calculate the speed of the dart when it hits the board.
To calculate the time of flight of the dart, we can use the vertical motion equation:
h = v₀y * t - 0.5 * g * t²
Where:
- h is the vertical displacement (1.73 m in this case)
- v₀y is the initial vertical velocity (which is the same as the vertical component of the initial velocity of the dart)
- g is the acceleration due to gravity (approximately 9.8 m/s²)
- t is the time of flight
Since the dart leaves your hand with a horizontal velocity and its vertical velocity is at a point 1.94 m above the ground, we can assume that the dart is launched from the ground level. Hence, the initial vertical displacement (h) is 0.
Therefore, the equation becomes:
0 = v₀y * t - 0.5 * g * t²
Next, let's calculate the initial vertical velocity (v₀y) using the equation:
v₀y = v₀ * sin(θ)
Where:
- v₀ is the initial speed of the dart
- θ is the angle of launch with respect to the horizontal plane
From the information given, we don't have the value of the launch angle (θ). However, since the dart hits the board at a height of 1.73 m and it was launched from a point 1.94 m above the ground, we can infer that the horizontal and vertical components of the initial velocity are equal.
So, we can use the Pythagorean theorem to find the initial speed (v₀):
v₀² = (v₀ * cos(θ))² + (v₀ * sin(θ))²
v₀ = √((v₀ * cos(θ))² + (v₀ * sin(θ))²)
v₀ = v₀ * √(cos²(θ) + sin²(θ))
v₀ = v₀
This means that the value of v₀ doesn't depend on the launch angle. Hence, we can calculate it independently.
Now, let's solve for the missing values in the equation:
0 = v₀y * t - 0.5 * g * t²
0 = v₀ * sin(θ) * t - 0.5 * g * t²
Since we don't know the value of θ, let's represent sin(θ) as a ratio:
sin(θ) = opposite/hypotenuse
sin(θ) = 1.73 m / 3.21 m
sin(θ) ≈ 0.5387
Now we can substitute the known values into the equation:
0 = v₀ * 0.5387 * t - 0.5 * 9.8 * t²
0 = 0.5387 * v₀ * t - 4.9 * t²
Now we have a quadratic equation in terms of time (t). To solve for t, we can use the quadratic formula:
t = (-b ± √(b² - 4ac)) / 2a
In this case, the equation is in the form of at² + bt + c = 0, where:
- a = -4.9
- b = 0.5387 * v₀ (we will solve for v₀ later)
- c = 0
Now let's calculate the time of flight (t) using the quadratic formula.
Please provide the value of the initial speed of the dart (v₀) to proceed with the calculation.