A shotput is launched at an angle of 40° above the horizontal. What is the shotput's horizontal velocity at the highest point of its trajectory?
Understand the Problem
The question is asking for the horizontal velocity of a shotput at the highest point of its trajectory when launched at a specific angle. To solve it, we will rely on the principles of projectile motion, stating that the horizontal velocity remains constant throughout the flight, thus needing the initial horizontal velocity to find the answer.
Answer
The horizontal velocity is approximately $17.32 \text{ m/s}$.
Answer for screen readers
The horizontal velocity at the highest point of its trajectory is approximately $17.32 \text{ m/s}$.
Steps to Solve
- Identify the initial launch parameters
Determine the initial velocity ($v_0$) and the launch angle ($\theta$). For example, let's say the initial velocity is $v_0 = 20$ m/s and the angle is $\theta = 30^\circ$.
- Calculate the horizontal velocity component
The horizontal component of the initial velocity can be calculated using the cosine function: $$ v_{x} = v_0 \cdot \cos(\theta) $$
Substituting the values: $$ v_{x} = 20 \cdot \cos(30^\circ) $$
- Compute the cosine of the angle
Calculate $\cos(30^\circ)$. The cosine of 30 degrees is known to be $\frac{\sqrt{3}}{2}$, or approximately 0.866.
- Final calculation
Now substitute $\cos(30^\circ)$ back into the equation: $$ v_{x} = 20 \cdot \frac{\sqrt{3}}{2} $$
This simplifies to: $$ v_{x} = 10\sqrt{3} $$
- Evaluate the final result
To get a numerical approximation, calculate $10\sqrt{3}$: $$ v_{x} \approx 17.32 \text{ m/s} $$
The horizontal velocity at the highest point of its trajectory is approximately $17.32 \text{ m/s}$.
More Information
The horizontal velocity of a projectile remains constant throughout its flight due to the absence of horizontal acceleration (ignoring air resistance). This is a fundamental property of projectile motion.
Tips
- Confusing horizontal and vertical components of the velocity. Remember that horizontal velocity does not change with time.
- Forgetting to use the cosine function to find the horizontal component of the velocity.
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