John was wondering at what speed an arrow leaves a bow when it is released. So he went to a field and shot the bow at an angle of 45 degrees. The arrow hit the ground at a distance of 600 feet. Ignore air resistance. For simplicity, assume that the arrow is released at an initial height of 0 feet.
The arrow leaves the bow at a speed of how many miles per hour?
the range for an angled shot is
R = v^2/g sin2θ
so, plug in your numbers. Remember to convert ft/s to mi/hr.
To calculate the speed at which the arrow leaves the bow, we can use the principles of projectile motion. The initial velocity of the arrow can be broken down into its horizontal and vertical components.
Given that the arrow is shot at an angle of 45 degrees, the initial velocity can be split equally between the horizontal and vertical components. This means that the initial velocity in the horizontal direction (Vx) is equal to the initial velocity in the vertical direction (Vy).
Since we're given the range (distance traveled in the horizontal direction) of the arrow, we can use the equation for horizontal motion to find the initial velocity (Vx):
Range = Vx * Time
In this case, the range is 600 feet, and the time of flight (Time) can be determined using the vertical motion equation:
Time = (2 * Vy) / g
Where g is the acceleration due to gravity (approximately 32.17 ft/s^2).
Knowing that the arrow hits the ground at a distance of 600 feet, we can use trigonometry to determine the vertical component of the initial velocity (Vy) by using the equation:
600 = Vy * Time
With these two equations, we can solve for Vx and Vy, allowing us to find the initial velocity.
Finally, we will convert the velocity from feet per second to miles per hour.
Let's now calculate the speed at which the arrow leaves the bow:
1. Calculate Time:
Time = (2 * Vy) / g
Time = (2 * Vy) / 32.17
2. Calculate Vy:
600 = Vy * Time
600 = Vy * [(2 * Vy) / 32.17]
600 = (2Vy^2) / 32.17
19,302 = 2Vy^2
Vy^2 = 19,302 / 2
Vy^2 = 9,651
Vy ≈ 98.2 ft/s
3. Calculate Vx:
Range = Vx * Time
600 = Vx * [(2 * Vy) / 32.17]
600 = Vx * (196.4 / 32.17)
600 = Vx * 6.11
Vx ≈ 98.2 ft/s
4. Calculate Initial Velocity:
V = sqrt(Vx^2 + Vy^2)
V = sqrt(98.2^2 + 98.2^2)
V ≈ sqrt(9,643 + 9,643)
V ≈ sqrt(19,286)
V ≈ 138.8 ft/s
5. Convert to miles per hour:
Speed (in miles per hour) = (138.8 ft/s) * (3600 seconds/hour) * (1 mile/5280 feet)
Speed ≈ 94.6 miles per hour
Therefore, the arrow leaves the bow at a speed of approximately 94.6 miles per hour.