A ball is shot from a compressed-air gun at twice its terminal speed.

a. What is the ball's initial acceleration, as a multiple of g, if it is shot straight up?

Question

A ball is shot from a compressed-air gun at twice its terminal speed.

a. What is the ball's initial acceleration, as a multiple of g, if it is shot straight up?

Accepted Answer

Hey, everyone. So this problem is working with drag forces and terminal velocity. Let's see what they're asking us, were asked to write the expression of the initial acceleration of a body of mass m thrown vertically upwards with an initial speed equal to triple its terminal speed express your answer as a multiple of the gravitational acceleration or G. Okay. So the first thing we can do here is draw a free body diagram. So when we have a mass thrown upward into the air, after it is released, the two forces that are acting on it are the drag force which acts in the direction opposite of the direction of motion and our weight. So from Newton's second law, we can recall that some of the forces this time are working in the Y direction equals mass times acceleration that some of our forces are gonna be minus F D or drag force minus weight equals mass times acceleration. And we can recall here that our drag force is given by the equation FD equals 1/ row C A B square borough is our density C is our drag coefficient A is the area and B is our velocity. They're telling us now that the initial speed Is equal to triple or three times it's terminal speed. And then we can recall that terminal speed is given by the equation square root of two M G times row C A. So from there, we can plug that back into our Newton's second law equation that looks like minus one half times row C A V squared is going to be three times BT, two mg over Rho C A squared, that's minus and then minus W equals N A. Our weight we can also recall is given by mass times gravity. So we'll simplify this out a little bit more ro C A and that's nine times two mg over Rho C A minus mg was um A. So from here, we see that a lot of things can cancel out row C A cancels here. We have a mass term that cancels in each of our terms And 1/2 year cancels with two. And so we are simply left with -9 G -G equals a Rewrite that as a equals -10 G. So that is the answer to this problem. We look at our multiple choice answers and that aligns with choice A. So A is the correct answer for this problem. That's all we have for this one. We'll see you in the next video.

A ball is shot from a compressed-air gun at twice its terminal speed. a. What is the ball's initial acceleration, as a multiple of g, if it is shot straight up?

Verified Solution

Key Concepts

Terminal Velocity

Acceleration due to Gravity (g)

Newton's Second Law of Motion