To solve this problem, we'll use the kinematic equations to determine the velocity components when the pea strikes the ceiling. After finding the horizontal and vertical components, we'll apply the Pythagorean theorem to calculate the total speed.

Given Data:

• Initial speed $v_0 = 7.31 \, \text{m/s}$
• Launch angle $\theta = 69.0^\circ$
• Height of the ceiling $y_f = 1.90 \, \text{m}$
• Gravitational acceleration $g = 9.81 \, \text{m/s}^2$

Step 1: Break down the initial velocity

We first find the horizontal and vertical components of the initial velocity using trigonometry:

$v_{0x} = v_0 \cdot \cos(\theta)$ $v_{0y} = v_0 \cdot \sin(\theta)$

Substituting the values:

$v_{0x} = 7.31 \cdot \cos(69^\circ) = 7.31 \cdot 0.3584 \approx 2.62 \, \text{m/s}$ $v_{0y} = 7.31 \cdot \sin(69^\circ) = 7.31 \cdot 0.9336 \approx 6.83 \, \text{m/s}$

Step 2: Use kinematic equations to find vertical velocity at the ceiling

Next, we apply the vertical motion equation to find the vertical velocity when the pea hits the ceiling. The equation is:

$v_y^2 = v_{0y}^2 - 2g \cdot (y_f - y_0)$

Where:

• $v_y$ is the vertical velocity at the ceiling
• $y_f = 1.90 \, \text{m}$ is the ceiling height
• $y_0 = 0 \, \text{m}$ (since the pea starts from the table)

Substituting the values:

$v_y^2 = (6.83)^2 - 2 \cdot 9.81 \cdot (1.90 - 0)$ $v_y^2 = 46.64 - 2 \cdot 9.81 \cdot 1.90$ $v_y^2 = 46.64 - 37.27$ $v_y^2 \approx 9.37$ $v_y = \sqrt{9.37} \approx 3.06 \, \text{m/s}$

Step 3: Use Pythagorean theorem to find the total speed

Now that we have both the horizontal and vertical components of velocity when the pea strikes the ceiling, we use the Pythagorean theorem to find the total speed:

$v = \sqrt{v_x^2 + v_y^2}$

Since the horizontal velocity $v_x = 2.62 \, \text{m/s}$ remains constant, we have:

$v = \sqrt{(2.62)^2 + (3.06)^2}$ $v = \sqrt{6.86 + 9.37}$ $v = \sqrt{16.23} \approx 4.03 \, \text{m/s}$

Final Answer:

The speed of the pea when it strikes the ceiling is approximately 4.03 m/s.

Do you want more details, or have any other questions?

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Here are five related questions to consider:

1. How long does it take for the pea to hit the ceiling?
2. What is the maximum height reached by the pea if the ceiling was not there?
3. What is the horizontal range of the pea if it wasn't stopped by the ceiling?
4. How would the pea's trajectory change if the launch angle was increased?
5. How much time does it take for the vertical velocity to reduce to zero?

Tip: The vertical velocity component changes due to gravity, but the horizontal velocity remains constant in projectile motion (without air resistance).