An archer standing on a cliff 48m above the level field below shoots an arrow at an angle of 30° above the horizontal with a speed of 80m/s. How far from the base of the cliff will the arrow land?
calculate the horizontal and vertical components of the arrow's flight
... horiz ... 80 m/s * cos(30º)
... vert ... 80 m/s * sin(30º)
plug the vert into the free-fall (height) equation to find the flight time
... h = 1/2 g t^2 + [80 m/s * sin(30º)] t + 48
... find t when h=0 , arrow on the ground
distance from cliff is the flight time multiplied by the horizontal velocity
h=78.09
To determine how far from the base of the cliff the arrow will land, we can break down the problem into horizontal and vertical components.
1. Horizontal Component:
The horizontal component of the arrow's initial velocity can be calculated by using the formula:
Horizontal component = Initial velocity × cos(angle)
Given that the initial velocity is 80 m/s and the angle is 30°, we can plug the values into the formula:
Horizontal component = 80 m/s × cos(30°)
Horizontal component ≈ 69.28 m/s
2. Vertical Component:
The vertical component of the arrow's initial velocity can be calculated by using the formula:
Vertical component = Initial velocity × sin(angle)
Given that the initial velocity is 80 m/s and the angle is 30°, we can plug the values into the formula:
Vertical component = 80 m/s × sin(30°)
Vertical component ≈ 40 m/s
3. Time of Flight:
The total time of flight can be calculated using the vertical component of the velocity and acceleration due to gravity:
Time of flight = (2 × Vertical component) / gravitational acceleration
Given that the vertical component is approximately 40 m/s, and the gravitational acceleration is approximately 9.8 m/s^2, we can plug the values into the formula:
Time of flight = (2 × 40 m/s) / 9.8 m/s^2
Time of flight ≈ 8.16 s
4. Distance:
The horizontal distance can be calculated by multiplying the horizontal component of velocity by the time of flight:
Distance = Horizontal component × Time of flight
Given that the horizontal component is approximately 69.28 m/s and the time of flight is approximately 8.16 s, we can plug the values into the formula:
Distance = 69.28 m/s × 8.16 s
Distance ≈ 565.97 m
So, the arrow will land approximately 565.97 m from the base of the cliff.
To determine the horizontal distance the arrow will travel before landing, we need to analyze the horizontal and vertical components of its motion separately.
First, let's determine the time it takes for the arrow to hit the ground using its vertical motion. We'll assume no air resistance.
The initial vertical velocity component (Vy) can be calculated using the given launch angle (30°) and the total initial velocity (80 m/s):
Vy = (80 m/s) * sin(30°)
Vy = 40 m/s * 0.5
Vy = 20 m/s
Using the equations of motion, we can calculate the time it takes for the arrow to reach the ground (t):
Vertical distance traveled (Δy) = 48 m (the height of the cliff)
Initial vertical velocity (Vy) = 20 m/s
Acceleration due to gravity (g) = 9.8 m/s²
Using the equation Δy = (Vy * t) + (0.5 * g * t²), we can rearrange it to solve for t:
48 m = (20 m/s * t) + (0.5 * 9.8 m/s² * t²)
Rearranging the equation, we get:
0.5 * 9.8 m/s² * t² + 20 m/s * t - 48 m = 0
This is a quadratic equation, which we can solve for t using the quadratic formula:
t = (-b ± sqrt(b² - 4ac)) / (2a)
For this equation, a = 0.5 * 9.8 m/s², b = 20 m/s, and c = -48 m. Plugging these values into the quadratic formula:
t = (-20 m/s ± sqrt((20 m/s)² - 4 * 0.5 * 9.8 m/s² * -48 m)) / (2 * 0.5 * 9.8 m/s²)
Solving this equation will give us the two possible values of t, but only the positive value makes physical sense since time cannot be negative.
Once we have the time it takes for the arrow to reach the ground, we can calculate the horizontal distance traveled using its horizontal motion:
Horizontal velocity component (Vx) = 80 m/s * cos(30°)
Horizontal distance traveled (S) = Vx * t
Plugging in the values and calculating, we find:
Vx = 80 m/s * cos(30°)
Vx = 80 m/s * 0.866
Vx ≈ 69.3 m/s
S = 69.3 m/s * t
Finally, substitute the positive value of t we obtained earlier to calculate the horizontal distance:
S = 69.3 m/s * t (positive value obtained)
Please solve the equation to find the exact value of time (t), then substitute it into the formula for horizontal distance (S) to determine the final result.