Find minimum initial velocity of object

[rajesan]

Homework Statement

Find the minimum initial velocity needed for a ball starting from the middle-top (highest point) of a semicircular mound with radius R to go over without touching the mound if the initial velocity has only a horizontal component.

Homework Equations

v=d/t

The Attempt at a Solution

this was a question on a test i just had. i didn't have time to prepare to well for this test so i have no idea how i did. i just used the formula v=d/t and substituted the variable R (radius) for d. then i solved fort which equaled t=R/v. then i plugged this value of t into the formula
"[tex]\Delta[/tex]d=vt+.5at[tex]^{2}[/tex]. with this i just solved for v and i got v=[tex]\sqrt{4.9R}[/tex]. I am pretty sure it is wrong and that i was supposed to use some formula for circular motion or something. could you please help me figure this question out?

 

[Delphi51]

You've got it!

 

[tomcue]

Your approach is on the right track, but there are a few issues with your solution. First, the formula v=d/t is for constant velocity, but in this problem, the velocity is changing due to gravity. So we need to use a different formula that takes into account acceleration.

Secondly, the formula you used, "\Deltad=vt+.5at^{2}, is for motion with constant acceleration, but in this problem, the acceleration is not constant (it changes as the ball moves along the semicircular mound).

To solve this problem, we can use conservation of energy. At the top of the mound, the ball has only potential energy (due to its height), and at the bottom of the mound, it has both potential and kinetic energy. We can set these two energies equal to each other and solve for the minimum initial velocity.

So the equation would be: mgh = 1/2mv^2, where m is the mass of the ball, g is the acceleration due to gravity, h is the height of the mound (which is equal to the radius R), and v is the initial velocity.

Solving for v, we get v = √(2gh) = √(2gR).

So the minimum initial velocity needed for the ball to go over the mound without touching it is √(2gR).

Note that this is the minimum velocity needed for the ball to just barely make it over the mound. If the initial velocity is slightly lower than this, the ball will touch the mound.