# What Value of h/H Maximizes Time to Reach the Ground?

• The legend
In summary, the problem is to find the value of h/H such that the total time for a body to fall from rest from height H, hit an inclined plane at height h, and reach the ground is maximized. Using projectile motion equations, we can solve for t1 and t2, the times it takes the object to reach H-h and h respectively. The final time is the sum of these two times. To maximize this time, we can use mathematical techniques such as differentiation. By equating t1+t2 to the time it takes for the object to fall from rest, we can solve for h/H and find that the maximum time is achieved when h/H=1/2.
The legend

## Homework Statement

A body falling from rest from given height H hits an inclined plane in its path at height. As a result of this impact the direction of velocity of the body becomes horizontal. For what value of h/H will body take max time to reach the ground?
ANS) h/H = 1/2

## Homework Equations

Newtons equations of motion
Projectile equations

## The Attempt at a Solution

v2 = u2 + 2as
v2= 2 * 10 * (H-h)

now time of flight of projectile from inclined plane
T = 2*v*sinx/g*cosy (x being angle of projection and y angle of incline)
this has to be equal to time of projectile dropped with no initial velocity
T = (2h/g)1/2

equating and substituting i get the answer to be
h/H = 4/5

(which is not the correct answer)

You want to find the ratio of h/H such that the total time the object falls is maximized. There are two parts to the problem; both deal with projectile motion.

Since we're dealing with time, we don't care about the x component of position, only the y. For the first part, we know that the body is falling from rest, and that the displacement is H-h. What equation can you use to solve for the time it takes the object to reach this point? (Call it t1.)

Then, the problem tells us that the object hits an inclined plane, changing the direction of the velocity to the horizontal. What does this tell us about $$v_{0x}$$ and $$v_{0y}$$? (Hint: set the displacement for the second part as h and solve for the time it takes to travel this distance, t2.)

Once you've solved for these two times, tfinal=t1+t2. What mathematical technique can you use to maximize a value with respect to a certain variable?

Last edited:
Oh Yeah! I got it!
Thanks a lot !

## 1. What is an inclined plane?

An inclined plane is a simple machine that is essentially a flat surface that is at an angle to the ground. It is commonly used to make it easier to move heavy objects to higher or lower surfaces.

## 2. How does an inclined plane work?

An inclined plane works by reducing the force needed to lift or move an object by increasing the distance over which that force is applied. This makes it easier to move heavy objects by spreading the force over a longer distance.

## 3. What is the formula for calculating the mechanical advantage of an inclined plane?

The formula for calculating the mechanical advantage of an inclined plane is MA = L/H, where MA is the mechanical advantage, L is the length of the inclined plane, and H is the height of the inclined plane.

## 4. What are some real-life examples of inclined planes?

Some real-life examples of inclined planes include ramps for wheelchairs, roads on hills, and slides at playgrounds. Inclined planes can also be found in everyday objects such as screwdrivers, axes, and stairs.

## 5. How does friction affect the use of inclined planes?

Friction can make it more difficult to use inclined planes because it acts against the applied force, making it harder to move objects. To reduce the effects of friction, lubricants or wheels can be used on the inclined plane surface.

Replies
18
Views
3K
Replies
7
Views
2K
Replies
7
Views
2K
Replies
13
Views
733
Replies
25
Views
819
Replies
8
Views
2K
Replies
34
Views
1K
Replies
3
Views
1K
Replies
10
Views
1K
Replies
8
Views
3K