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Let's solve the problem step by step:

### Given Data:
- Initial speed \( u = 29.4 \, \text{m/s} \)
- Final speed \( v = 14.7 \, \text{m/s} \)
- Acceleration due to gravity \( g = -9.8 \, \text{m/s}^2 \) (since the motion is vertical and against gravity, the acceleration is negative)
- The ball is projected upwards and we are asked to find the distance traveled until the speed reduces to \( 14.7 \, \text{m/s} \).

We can use the following **equation of motion**:
\[
v^2 = u^2 + 2 a s
\]
where:
- \( v \) is the final velocity,
- \( u \) is the initial velocity,
- \( a \) is the acceleration (which is \( -9.8 \, \text{m/s}^2 \)),
- \( s \) is the distance traveled.

### Step 1: Apply the formula
\[
(14.7)^2 = (29.4)^2 + 2(-9.8) s
\]

Now let's solve for \( s \). I'll calculate this.The distance traveled by the ball while its speed decreases from \( 29.4 \, \text{m/s} \) to \( 14.7 \, \text{m/s} \) is approximately \( 33.08 \, \text{m} \).

Would you like further explanation on any part of this solution?

Here are 5 related questions to expand your understanding:
1. How would the solution change if the ball were projected from ground level instead of 19.6 m above?
2. How much time does the ball take to reach the speed of 14.7 m/s?
3. What is the maximum height the ball reaches?
4. How would air resistance affect the motion of the ball in this scenario?
5. How far will the ball fall after it stops moving upwards?

**Tip:** Always ensure that signs (positive or negative) for acceleration match the direction of motion in such problems.

A small ball is projected vertically upwards with speed 29.4 m/s from a point A which is 19.6 m above horizontal ground. The ball is modeled as a particle moving freely under gravity until it hits the ground. It is assumed that the ball does not rebound. Find the distance travelled by the ball while its speed is less than 14.7 m/s.

Solve the kinematics problem of a ball projected vertically upwards with an initial speed of 29.4 m/s. Learn how to apply the equation of motion v^2 = u^2 + 2as to find the distance traveled as the speed decreases to 14.7 m/s. Ideal for students in Grades 10-12.

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