An object that has a mass of *m* kg was thrown straight up from a platform which is *h* m above the ground. The initial velocity was

The air resistance is proportional to the velocity of the object and is equal to *av* N. Write down the equation for the distance traveled by the object at any time. Find the velocity of the object at the ground level for

**Note:** assume the object does not hit the platform when falling down.

Firstly, two cases should be considered:

- The object moving up.
- The object falling down.

The free body diagram for it will be as follows:

Let’s take downward to be a positive direction.

Knowing that the acceleration is the first derivative of the speed, let’s write down the basic equation for both cases (the resistance force is always opposite to the vector of velocity):

Initial Values:

Up:

Where

is the time at which the object will reach the highest point.

Solving the equation:

To find the velocity near the ground level, we need to determine the formula for the path traveled:

Now, we should find the time when the object falls on the ground. Let’s simplify the above equation:

Now, the following equation should be solved:

Knowing the exact values, the solution to the above equation can be found through iterative methods:

The value of *t1* is not the correct value, as the time cannot be negative. So, *t2* is the answer.

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