Question

How do I find the initial speed of a shot put launched from 6.00 feet above the ground at an angle of 42.0 degrees?

Answer 1

To find the initial speed required for a shot put launched from a height of 6.00 feet at an angle of 42.0 degrees, we can use the physics of projectile motion. For this example, we will need to know the target horizontal distance the shot put is supposed to travel (the range) and the height from which it is launched.

The basic equations of projectile motion are:

1. **Horizontal motion**:
\[
x = v_0 \cdot \cos(\theta) \cdot t
\]
where:
- \(x\) = horizontal distance (range)
- \(v_0\) = initial speed
- \(\theta\) = launch angle (42.0 degrees)
- \(t\) = time of flight

2. **Vertical motion**:
\[
y = h + v_0 \cdot \sin(\theta) \cdot t - \frac{1}{2} g t^2
\]
where:
- \(y\) = final vertical position (which will be 0 when hitting the ground)
- \(h\) = initial height (6.00 feet)
- \(g\) = acceleration due to gravity (approximately 32.2 feet/s²)

### Step-by-Step Procedure

1. **Set up coordinates**:
We choose the origin just at the launch height of 6.00 feet. Thus, the final vertical position \(y\) should be -6.00 feet when the projectile lands (since it starts at 6.00 feet and we consider down as negative).

2. **Write the vertical motion equation** and replace \(y\) with -6.00:
\[
-6 = 6 + v_0 \cdot \sin(42^\circ) \cdot t - \frac{1}{2} g t^2
\]

3. **Simplify**:
\[
0 = v_0 \cdot \sin(42^\circ) \cdot t - \frac{1}{2} g t^2 + 12
\]

4. **Use the horizontal motion equation** to relate \(v_0\) and \(t\):
Rearranging, we find that:
\[
t = \frac{x}{v_0 \cdot \cos(42^\circ)}
\]

5. **Substituting for \(t\)**:
We can now substitute \(t\) back into the vertical equation:
\[
0 = v_0 \cdot \sin(42^\circ) \cdot \left(\frac{x}{v_0 \cdot \cos(42^\circ)}\right) - \frac{1}{2} g \left(\frac{x}{v_0 \cdot \cos(42^\circ)}\right)^2 + 12
\]

This will give you an equation in terms of \(v_0\) and \(x\).

6. **Rearranging for \(v_0\)** will give you the initial speed needed to hit the target.

### Additional Information Needed
You need to provide the horizontal distance \(x\) where the shot put is intended to land in order to solve for \(v_0\).

### Conclusion
In summary, to find the initial speed, you'll plug in the horizontal distance into the derived equations and solve for the required initial velocity \(v_0\). The range of the shot put will affect the initial velocity required to reach the target.