an archer standing on a cliff 48m above the level field below shoots an arrow at an angle of 30• above horizontal with a speed of 80m/s.how far from the base of the cliff will the arrow land.
To find the horizontal distance traveled by the arrow, we need to break down the initial velocity into horizontal and vertical components. The horizontal component of the initial velocity remains constant throughout the flight because there is no horizontal acceleration.
Given:
Height of the cliff (h) = 48m
Angle above the horizontal (θ) = 30°
Initial velocity (V) = 80m/s
Step 1: Determine the horizontal and vertical components of the initial velocity.
The horizontal component (Vx) can be calculated using the formula: Vx = V * cos(θ)
Vx = 80m/s * cos(30°)
Vx = 80m/s * √3/2
Vx = 80 * √3 / 2
Vx = 80 * 1.73 / 2
Vx ≈ 69.28m/s
The vertical component (Vy) can be calculated using the formula: Vy = V * sin(θ)
Vy = 80m/s * sin(30°)
Vy = 80m/s * 1/2
Vy = 40m/s
Step 2: Calculate the time of flight.
The time of flight (t) can be determined using the vertical component (Vy) and gravity (g), which is approximately equal to 9.8 m/s².
h = Vy * t - (1/2) * g * t² (where h is the height and g is the acceleration due to gravity)
48 = 40t - 4.9t²
4.9t² - 40t + 48 = 0
Solving the above quadratic equation, we get two possible values for t:
t = 2.857s or t = 3.333s (approximately)
Since the time of flight cannot be negative, we take the positive value: t ≈ 3.333s
Step 3: Calculate the horizontal distance traveled (d).
The horizontal distance traveled (d) is equal to the horizontal component of the initial velocity (Vx) multiplied by the time of flight (t).
d = Vx * t
d ≈ 69.28m/s * 3.333s
d ≈ 231.04m
Therefore, the arrow will land approximately 231.04 meters from the base of the cliff.
To find the horizontal distance that the arrow will travel before it lands, we need to use the initial velocity of the arrow and the angle at which it is shot.
Step 1: Split the initial velocity into horizontal and vertical components:
The horizontal component of velocity (Vx) remains constant throughout the motion, while the vertical component of velocity (Vy) changes due to the effect of gravity.
Given:
Initial velocity (V) = 80 m/s
Launch angle (θ) = 30 degrees
Using trigonometric functions, we can find the horizontal and vertical components of velocity:
Vx = V * cos(θ)
Vy = V * sin(θ)
Vx = 80 * cos(30)
Vx ≈ 80 * 0.866
Vx ≈ 69.28 m/s
Vy = 80 * sin(30)
Vy ≈ 80 * 0.5
Vy ≈ 40 m/s
Step 2: Calculate the time of flight:
The time it takes for the arrow to reach the ground can be determined by analyzing the vertical motion.
Using the formula for vertical displacement under constant acceleration:
Vy = g * t
where g is the acceleration due to gravity (approximately 9.8 m/s²) and t is the time of flight.
40 = 9.8 * t
t ≈ 40 / 9.8
t ≈ 4.08 s
Step 3: Calculate the horizontal distance traveled:
To find the distance traveled horizontally, multiply the horizontal component of velocity by the time of flight.
Distance = Vx * t
Distance ≈ 69.28 m/s * 4.08 s
Distance ≈ 283.14 m
Therefore, the arrow will land approximately 283.14 meters from the base of the cliff.
D=V costheta ×t
D=80m/s cos30 ×0.6927
D=48.0m
Range = Vo^2*sin(2A)/g.
Vo = 80 m/s, A = 30o, g = 9.8 m/s^2.