To solve this problem, we need to analyze the motion of both the arrow and the apple. The arrow is shot at a velocity of 30 m/s at an angle of 20 degrees, and we need to determine the initial speed of the apple that is thrown straight up so that it meets the arrow at a distance of 30 m downrange. Additionally, we will find out when the apple should be thrown after the arrow is launched.
Breaking Down the Arrow's Motion
The first step is to determine the horizontal and vertical components of the arrow's velocity. We can use trigonometric functions for this:
- Horizontal velocity (Vx): Vx = V * cos(θ) = 30 m/s * cos(20°)
- Vertical velocity (Vy): Vy = V * sin(θ) = 30 m/s * sin(20°)
Calculating these values:
- Vx = 30 * cos(20°) ≈ 30 * 0.9397 ≈ 28.19 m/s
- Vy = 30 * sin(20°) ≈ 30 * 0.3420 ≈ 10.26 m/s
Time of Flight to Reach 30 m
Next, we need to find the time it takes for the arrow to travel 30 m horizontally. The horizontal motion can be described by the equation:
Distance = Velocity × Time
Rearranging gives us:
Time = Distance / Velocity = 30 m / 28.19 m/s ≈ 1.06 seconds
Vertical Motion of the Arrow
Now, we need to find the vertical position of the arrow at this time. The vertical motion of the arrow can be described by the equation:
y = Vy * t - (1/2) * g * t²
Where g is the acceleration due to gravity (approximately 9.81 m/s²). Plugging in the values:
y = 10.26 m/s * 1.06 s - (1/2) * 9.81 m/s² * (1.06 s)²
Calculating this gives:
- y ≈ 10.26 * 1.06 - 0.5 * 9.81 * 1.1236
- y ≈ 10.89 - 5.51 ≈ 5.38 m
Apple's Motion
The apple is thrown straight up and must reach the same height (5.38 m) at the same time (1.06 seconds). The motion of the apple can be described by the equation:
y = V0 * t - (1/2) * g * t²
Setting y to 5.38 m and solving for V0 gives us:
5.38 = V0 * 1.06 - (1/2) * 9.81 * (1.06)²
Rearranging this equation:
V0 * 1.06 = 5.38 + 5.51
V0 * 1.06 = 10.89
V0 = 10.89 / 1.06 ≈ 10.27 m/s
Timing of the Apple Throw
Since the apple must be thrown just before the arrow reaches it, we need to calculate when to throw the apple. The apple should be thrown 1.06 seconds after the arrow is launched to meet it at the height of 5.38 m.
Summary of Results
- Initial speed of the apple: Approximately 10.27 m/s
- Time after the arrow is launched to throw the apple: 1.06 seconds
By understanding the motion of both the arrow and the apple, we can effectively determine the necessary conditions for them to meet in mid-air. This problem illustrates the principles of projectile motion and the interplay between horizontal and vertical movements.
