An archer shoots an arrow that leaves the bow at an angle of 42 degrees above the horizontal. The arrow hits a target located 135 m away and 23 m above the height from which the arrow was shot. Given that the mass of the arrow is .125 kg and that it remained in contact with the string 1.28 m, what was the force applied to the arrow by the string of the bow?
Can someone at least tell me where to begin? I'm lost when it comes to these types of problems.
To solve this problem, we can use the principles of projectile motion and the equations of motion. The force applied to the arrow by the string of the bow can be calculated using the laws of physics.
1. First, we need to break down the given information and identify the variables provided:
- Angle of projection (θ): 42 degrees
- Distance traveled (horizontal displacement, x): 135 m
- Vertical displacement (y): 23 m
- Mass of the arrow (m): 0.125 kg
- Distance traveled while in contact with the string (s): 1.28 m
2. We can use the equations of projectile motion to find the initial velocity of the arrow and the time of flight. Let's start by finding the initial velocity (v):
- The horizontal component of the velocity (vx) can be found using the equation: vx = v * cos(θ)
- The vertical component of the velocity (vy) can be found using the equation: vy = v * sin(θ)
3. To find the initial velocity (v), we will use the equation for the vertical displacement (y): y = vyt - (1/2) * g * t^2
- Rearranging the equation, we have: t = (√(2y/g))
4. We can now substitute the value of time (t) into the equation for horizontal displacement (x): x = vxt
- Rearranging the equation, we have: v = x / (t * cos(θ))
5. Now that we have found the initial velocity (v), we can calculate the acceleration (a) using the equation: a = v / t
- Substituting the values, we have: a = v / (√(2y / g))
6. Finally, we can calculate the force applied to the arrow by the string using Newton's second law: F = m * a
By following these steps and substituting the given values into the appropriate equations, you can find the force applied to the arrow by the string of the bow.