Calculating Initial Velocity for Vertical Launch

The problem asks for the initial velocity ($v_0$) of an object launched vertically upwards, given its maximum height ($h_{max}$) and the acceleration due to gravity ($g$).

1. Unit Conversion

First, convert the maximum height from centimeters to meters:

• $h_{max} = 1000 \text{ cm} = \frac{1000}{100} \text{ m} = 10 \text{ m}$

2. Kinematic Analysis

At the maximum height, the object's instantaneous velocity ($v$) is 0 m/s. We can use the following kinematic equation:

$ v^2 = v_0^2 + 2 a \Delta y $

• Final velocity, $v = 0 \text{ m/s}$
• Initial velocity, $v_0 = ?$
• Acceleration, $a = -g = -9.8 \text{ m/s}^2$ (negative because gravity acts downwards)
• Displacement, $\Delta y = h_{max} = 10 \text{ m}$

3. Solving for Initial Velocity

Substitute the known values into the equation:

$ (0 \text{ m/s})^2 = v_0^2 + 2 (-9.8 \text{ m/s}^2) (10 \text{ m}) $

Simplify the equation:

$ 0 = v_0^2 - 196 \text{ m}^2/\text{s}^2 $

Rearrange to solve for $v_0^2$:

$ v_0^2 = 196 \text{ m}^2/\text{s}^2 $

Take the square root to find $v_0$:

$ v_0 = \sqrt{196 \text{ m}^2/\text{s}^2} $

$ v_0 = 14 \text{ m/s} $

Conclusion

The initial velocity of the object was 14 m/s.