Terminal velocity of a ping pong ball

[disruptors]

Hey ppl, just a quick question..

You have a 10 g ping pong ball with a diameter of 3.5 cm...
a)The ball is shot straight up at twice its terminal speed. What is it’s
initial acceleration?

i know D=W at terminal speed but when u double it do you just assume D=4D and if you do how come?

The answer is a=-5g as summation of Fy=-D-w=-4w-w=-5mg=ma

the -4w is just throwing me off...

Thanks

 

[Tide]

If $v = 2 v_t$ then $v^2 = 4 v_t^2$ which accounts for your factor of 4.

 

[wais]

for your question! To understand the concept of terminal velocity, we need to first understand what it means. Terminal velocity is the maximum velocity that an object can reach when falling through a fluid (such as air) due to the opposing force of drag. When an object is initially dropped, it will accelerate due to the force of gravity. However, as it falls, the force of drag increases until it is equal to the force of gravity, resulting in a constant velocity known as terminal velocity.

In the case of a ping pong ball, its terminal velocity is reached when the upward force of drag (D) is equal to the downward force of gravity (W). Doubling the terminal speed means that the velocity has increased, but the forces of drag and gravity are still equal. Therefore, the initial acceleration would still be the same as the acceleration due to gravity, which is -g or approximately -9.8 m/s^2.

To better understand the equation provided in the question, let's break it down. The equation for summation of forces in the vertical direction is Fy = -D - W = -4w - w. This means that the total force in the vertical direction (Fy) is equal to the sum of the force of drag (-D) and the weight of the ball (-W). Since the ball is now moving at twice its terminal speed, the force of drag is now equal to -4w (four times the original drag force), while the weight remains the same at -w. Therefore, the total force in the vertical direction is now -5w, which is equal to the mass (m) of the ball multiplied by the acceleration (a) (-5mg = ma). This results in an acceleration of -5g or -5 * 9.8 m/s^2 = -49 m/s^2.

I hope this helps clarify your confusion about the -4w in the equation. Keep in mind that this is just a simplified explanation and in reality, there are other factors that can affect the terminal velocity of a ping pong ball, such as air density and surface roughness. It's always important to double check your calculations and make sure all units are consistent when solving physics problems. Good luck with your studies!