n the figure, a stone is projected at a cliff of height h with an initial speed of 46.0 m/s directed at an angle θ 0 = 50.0° above the horizontal. The stone strikes at A, 5.19 s after launching. Find (a) the height h of the cliff, (b) the speed of the stone just before impact at A, and (c) the maximum height H reached above the ground.

To solve this problem, we need to break down the motion of the stone into its horizontal and vertical components. The stone is projected with an initial speed and at an angle, which means we can use some basic physics principles to find the height of the cliff, the speed just before impact, and the maximum height reached. Let’s tackle each part step by step.

Calculating the Height of the Cliff (h)

First, we need to determine the vertical motion of the stone. The initial vertical velocity can be calculated using the formula:

• Initial vertical velocity (v_0y) = v₀ * sin(θ)

Where:

• v₀ = 46.0 m/s (initial speed)
• θ = 50.0° (angle of projection)

Calculating this gives:

• v_0y = 46.0 m/s * sin(50.0°) ≈ 46.0 m/s * 0.7660 ≈ 35.2 m/s

Next, we can use the vertical motion equation to find the height of the cliff:

• h = v_0y * t - 0.5 * g * t²

Where:

• t = 5.19 s (time of flight)
• g = 9.81 m/s² (acceleration due to gravity)

Plugging in the values:

• h = 35.2 m/s * 5.19 s - 0.5 * 9.81 m/s² * (5.19 s)²
• h = 182.4 m - 0.5 * 9.81 m/s² * 26.9 s²
• h = 182.4 m - 132.5 m ≈ 49.9 m

Finding the Speed Just Before Impact (v_A)

To find the speed of the stone just before it strikes the cliff, we need to calculate both the horizontal and vertical components of the velocity at that moment. The horizontal component remains constant:

• v_x = v₀ * cos(θ)

Calculating the horizontal component:

• v_x = 46.0 m/s * cos(50.0°) ≈ 46.0 m/s * 0.6428 ≈ 29.6 m/s

For the vertical component just before impact, we use the equation:

• v_y = v_0y - g * t

Calculating the vertical component:

• v_y = 35.2 m/s - 9.81 m/s² * 5.19 s
• v_y = 35.2 m/s - 51.0 m/s ≈ -15.8 m/s

The negative sign indicates that the stone is moving downward. Now, we can find the resultant speed just before impact using the Pythagorean theorem:

• v_A = √(v_x² + v_y²)

Calculating this gives:

• v_A = √((29.6 m/s)² + (-15.8 m/s)²)
• v_A = √(876.16 + 249.64) ≈ √1125.8 ≈ 33.5 m/s

Determining the Maximum Height (H)

To find the maximum height above the ground, we first calculate the maximum height above the launch point using the formula:

• H = (v_0y)² / (2g)

Substituting the values:

• H = (35.2 m/s)² / (2 * 9.81 m/s²)
• H = 1245.44 m²/s² / 19.62 m/s² ≈ 63.6 m

Finally, to find the total height above the ground, we add the height of the cliff:

• Total height = H + h

Calculating this gives:

• Total height = 63.6 m + 49.9 m ≈ 113.5 m

In summary, we have:

• (a) The height of the cliff (h) is approximately 49.9 m.
• (b) The speed of the stone just before impact (v_A) is approximately 33.5 m/s.
• (c) The maximum height (H) above the ground is approximately 113.5 m.