### What will happen to a ball thrown vertically upwards?

##### April 7, 2017

## Motion of a ball thrown vertically upwards

You must have seen someone throwing a ball vertically upwards or probably you have thrown it yourself. Yes, the ball goes up for a while and then stops and starts to come down and touches the ground with a bang if you can’t catch it. Let’s discuss the phases of this traversal and motion with some formula and examples. Certainly this motion is due to the earth’s gravity on the ball.Here we will discuss about the acceleration of the ball thrown vertically upwards – both during its upward and downward motion. We will also consider the maximum height reached and the time taken for upward and downward movements. So lets start.

**Upward movement of the ball**

You throw the ball vertically upwards with some velocity say **v1**, which we will consider as the initial velocity for the upward path.

After certain time period** **say** t,** the ball reaches a height beyond which it can’t move upwards anymore and stops there i.e.** its velocity becomes zero at that height.**

The height where the velocity becomes zero which is the **maximum height** the ball went upward, say is **H**. And for this upward movement, the **final velocity v2 is 0** because the ball has stopped at the end of this upward traversal.

##### acceleration of a ball thrown vertically upwards-during upward movement

It’s pretty evident that after the upward throw, the velocity of the ball gradually decreases i.e. a negative acceleration was working on the ball.

The acceleration is negative because this acceleration is directing downwards while the ball is moving upward and because of this negative acceleration the velocity of the ball is gradually decreasing.

And Yes. this acceleration is nothing but the **acceleration due to the earth’s gravity** (or gravitational pull or force) on the ball. It’s value is generally taken as 9.8 m/s^2. See here how acceleration due to gravity varies with height and depth wrt the surface of the earth.

##### time taken by the upward movement

As this **acceleration (g)** is working opposite to the upward velocity we have to use a negative sign in the formula below, used for the upward movement of the ball. We know the value of g in SI is 9.8 m/second square.

Using one of the equations of motion,

**v2 = v1 – gt ……………………(i)**

**As v2=0**, (at the highest point the velocity becomes zero), then we can write the previous equation as follows:

** 0 = v1 – gt**

or, **t = v1/g **…………………….(ii)

So from equation (ii), the **time taken by the ball to reach the maximum height** is = (Initial Velocity with which the ball is thrown vertically upwards) / (acceleration due to gravity)…..(iii)

So just for example, if a ball is thrown vertically upwards with 98 m/s velocity, then to reach the maximum height it will take = 98/9.8 =10 seconds.

##### maximum height reached during upward movement

And the maximum height H reached is obtained from the formula:

**v2 ^{2}=v1^{2}-2gH ( under negative acceleration) ……………… (iii)**

As v2=0, (**at the highest point the velocity becomes zero**), so we can rewrite equation iii as:

0 = v1^{2 }– 2gH

**or H= v1 ^{2}/2g …………(iv) (equation of maximum height) …………… (iv)**

Therefore if a ball is thrown vertically upwards with 98 m/s velocity, the maximum height reached by it would be = (98 x98 )/(2 x 9.8) meter = 490 meter.

###### What are the velocity and acceleration of the ball when it reaches the highest point?

Just summing up the answer here though we have already gone through this in the section above.

The **velocity at the highest point** is zero as the ball momentarily halts there before starting its downward movement.

And the **acceleration working on the ball at this point** is the acceleration due to gravity (g) and this time it’s considered positive i.e. downwards as the ball is now ready to free fall. (ignoring air resistance)

**Downward movement of the ball**

##### acceleration of a ball thrown vertically upwards-during downward movement

As the ball reaches the maximum height now it starts its** free fall **towards the earth. The force applied on it is again the gravity and this time its having a **positive acceleration** i.e. its following the same direction of the **acceleration due to gravity (g)**

##### time for the downward movement

It can be proved that the time for the downward movement or the time taken by the ball to fall from the highest point and reach the ground is same as the time required for the upward movement = v1/g.

###### Total time required for the upward and downward movement

So for a vertically thrown object the total time taken for its upward and downward movements is **2v1/g**

###### velocity of the ball *just before touching the ground
*

**And one important point**, during the downward fall the magnitude of the velocity of the ball *just before touching the ground* would be same as the magnitude of the velocity with which it was thrown upwards (v1 here).

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