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This question involves analyzing the motion and collision of a dart and a block, where the dart is launched at an angle and sticks to a wooden block. Let’s break down each part:

### Part (a): **Speed of the dart just before striking the block**
From the notes:
- The dart is launched at 30° with an initial speed of 10 m/s.
- The formula for kinetic energy conservation: \(\frac{1}{2} m v^2 = \frac{1}{2} m \cos(30^\circ) v\).

Substituting values:
\[
\frac{1}{2}(0.02)\cos(30^\circ)(10)^2 \implies v = 0.0866 \text{ m/s}.
\]
(However, this small velocity seems unusual, so we should clarify that the dart's speed might refer to horizontal speed post-launch).

### Part (b): **Horizontal distance \(d\) between the launching point and point directly below the block**
The formula used here:
\[
d = \frac{v_0^2}{g} \sin(2 \theta).
\]
For the dart:
\[
d = \frac{10^2}{9.8} \sin(2 \times 30^\circ) = 88.37 \text{ m}.
\]
This represents the range of the projectile motion.

### Part (c): **Speed of the block just after the dart strikes**
Conservation of momentum is used:
\[
m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f.
\]
Given:
- \(m_1 = 0.02 \, \text{kg}\) (dart),
- \(m_2 = 0.10 \, \text{kg}\) (block),
- The initial dart speed is \(v_1 = 10 \, \text{m/s}\),
- The block is initially at rest \(v_2 = 0\).

This gives:
\[
0.02(10) + 0.1(0) = (0.02 + 0.1) v_f,
\]
\[
0.2 = 0.12 v_f \implies v_f = \frac{0.2}{0.12} = 1.67 \, \text{m/s}.
\]
The speed of the block and dart system just after collision is approximately 1.67 m/s.

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### Would you like further clarification on any part of this solution or more detailed steps?

Here are some related questions:
1. What assumptions are made in projectile motion problems?
2. How does the angle of projection affect the range of a projectile?
3. What is the significance of using conservation of momentum in collisions?
4. How would air resistance alter the results of this problem?
5. Can you calculate the height at which the dart was launched?

**Tip:** In problems involving angles, always decompose the initial velocity into horizontal and vertical components to simplify calculations.

A small dart of mass 0.020 kg is launched at 30° above the horizontal at an initial speed of 10 m/s. At its highest point, it collides with and sticks to a wooden block of mass 0.10 kg suspended by a string. The task involves determining the speed of the dart before impact, the horizontal distance covered by the dart, and the speed of the block after collision.

Solve a physics problem involving the collision of a dart and a block. The problem covers concepts like projectile motion and conservation of momentum, using trigonometry and kinematic equations to calculate speeds and distances. Suitable for high school students in Grades 11-12.

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