Question

A ball is rolled off the edge of a table with a horizontal velocity of 3.0 ft/s. What will be the ball's velocity 0.10 s later?

Answer 1

The final vertical velocity is given as follows.\[v_{y,f} = \underbrace{v_{y,i}}_0 + a_{y}t \Rightarrow v_{y,f} = -gt\]The horizontal velocity \(v_x\) is invariant with respect to time. The final velocity is given by the Pythagorean theorem.\[v=\sqrt{v_x^2+v_{y,f}^2} \Rightarrow \boxed{v=\sqrt{v_x^2+(gt)^2}}\]

Answer 2

Let's look at the picture to get an understanding of what's going on. We know that it initially is going at 3.0ft/s to the right (according to our picture here) at location (A). There is no downwards velocity because it's on the table still and gravity can't accelerate it downwards. There is never any change in velocity in the x-direction because there is no gravity or wind to blow it faster or slower, so due to momentum it will keep going at the same speed to the right.

After it falls off the table, it will start to accelerate downwards due to gravity. Simply calculate the velocity downwards in the y-direction due to gravity after 10 seconds with your standard formula \[V _{y final} =V _{y initial} at\] Where you know that your initial velocity in the y-direction is 0ft/s and your acceleration is 32.2ft/s^2 and from the probelm we want to know where it is after 10 seconds.

Awesome, now we have the velocity in the x-direction (since velocity is constant in x-direction it's going to be 3.0ft/s for all time) and the y-direction's velocity from what we just calculated.



So how do you get your total velocity? Well, that's just a triangle when you look at it, where the total velocity is the hypotenuse of the x and y velocities, so we can apply the pythagorean theorem to it and find your velocity! Make sense?