While playing lawn darts with his friend, Jack notices that when he throws a dart up at 60 degrees it travels 11 meters. How high did the dart travel
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To find out how high the dart traveled, we need to use some basic trigonometry.
First, we'll break down the motion of the dart into its vertical and horizontal components. The vertical component determines the height achieved by the dart, while the horizontal component determines the distance traveled.
In this case, we know the distance traveled (11 meters) and the angle at which the dart was thrown (60 degrees). Since we're interested in the height, we'll focus on the vertical component.
The vertical component of the motion can be found using the formula:
Vertical distance = Initial velocity * time * sin(angle)
In this case, the initial velocity is the speed at which the dart was thrown upwards, and the time is the total time the dart was in the air.
However, we are not given the initial velocity or time directly. We can still work it out by using the horizontal component.
The horizontal component of the motion can be found using the formula:
Horizontal distance = Initial velocity * time * cos(angle)
We know the horizontal distance (11 meters), and we can find the time using this formula. Rearranging the equation, we have:
time = Horizontal distance / (Initial velocity * cos(angle))
Now that we have the time, we can substitute it into the formula for vertical distance to find the height:
Vertical distance = (Initial velocity * (Horizontal distance / (Initial velocity * cos(angle)))) * sin(angle)
Simplifying this equation, we can cancel out the terms "Initial velocity" and solve for the vertical distance:
Vertical distance = Horizontal distance * tan(angle)
Substituting the given values, we have:
Vertical distance = 11 * tan(60)
Calculating this, we get:
Vertical distance = 11 * 1.732
Thus, the height the dart traveled is approximately 19.05 meters.
To find out how high the dart traveled, we need to break down the motion of the dart into horizontal and vertical components.
First, let's consider the vertical motion of the dart. We know that the dart was thrown at an angle of 60 degrees and traveled a horizontal distance of 11 meters. Assuming no air resistance, we can use the equation for vertical motion to determine the height.
The equation for vertical motion is given by:
h = (v^2 * sin^2(theta)) / (2 * g)
Where:
h is the height
v is the initial velocity
theta is the angle of projection
g is the acceleration due to gravity (approximately 9.8 m/s^2)
In this case, we know the angle of projection (theta) is 60 degrees. Let's assume the initial velocity (v) of the dart is unknown.
To find the height, we need to find the initial velocity (v). Since the dart travels 11 meters horizontally, we can use the horizontal component of the initial velocity (v_x) to find the time of flight (t) using the equation:
d = v_x * t
Where:
d is the horizontal distance (11 meters in this case)
v_x is the horizontal component of the initial velocity
The horizontal component of the initial velocity can be found using the equation:
v_x = v * cos(theta)
Where:
cos(theta) is the cosine of the angle of projection
Now, let's solve for v_x:
v_x = v * cos(theta)
Since cos(60 degrees) = 0.5:
v_x = v * 0.5
Next, we can substitute v_x into the equation for the time of flight:
d = v_x * t
11 = v * 0.5 * t
Now, let's solve for t:
t = (11 * 2) / v
The time of flight (t) can be substituted into the equation for the height (h):
h = (v^2 * sin^2(theta)) / (2 * g)
h = (v^2 * sin^2(60 degrees)) / (2 * 9.8)
Now, our equation for the height (h) involves both the initial velocity (v) and the height itself. We can solve this equation using trial and error or numerical methods to find the value of h.