What is the initial vertical velocity of the t-shirt (in m/s)?

Understand the Problem

The question is asking for the initial vertical velocity of a t-shirt launched at an angle, using the provided initial velocity of 50 m/s and an angle of 40 degrees. We will use trigonometric functions to determine the vertical component of the initial velocity.

Answer

The initial vertical velocity of the t-shirt is approximately $32.14 \text{ m/s}$.

Steps to Solve

Identify the components of velocity

To find the initial vertical velocity component ($v_{y}$), we can use the trigonometric sine function since it relates to the angle of launch.

Use the sine function

The formula for the vertical component of the velocity is:

$$ v_{y} = v \cdot \sin(\theta) $$

Where:

$v$ is the initial velocity (50 m/s)
$\theta$ is the angle of launch (40 degrees)

Substitute the values

Now we substitute the given values into the formula:

$$ v_{y} = 50 \cdot \sin(40^\circ) $$

Calculate the sine value

First, calculate $\sin(40^\circ)$. This value is approximately $0.6428$.

Complete the calculation

Now calculate:

$$ v_{y} = 50 \cdot 0.6428 $$

This results in:

$$ v_{y} \approx 32.14 \text{ m/s} $$

The initial vertical velocity of the t-shirt is approximately $32.14 \text{ m/s}$.

More Information

The vertical velocity component helps determine how high the object will rise before falling back down, a key aspect of projectile motion. Understanding this concept is vital for anyone studying physics or engineering.

Tips

Confusing angle types: Sometimes, students mix up degrees and radians when calculating the sine of an angle. Always ensure your calculator is set to the correct mode.
Forgetting the sine function: Some might mistakenly try to use cosine instead of sine for vertical components. Remember, the vertical component uses $\sin$.

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