# Freefall questions

If one ball was thrown down with an initial velocity and another ball was thrown up with the same initial velocity. a) Which would have greater speed when it reaches the ground? b)Which has a greater displacement? c) Which ball will hit the ground first? d) Which travelled a greater distance?

I think they would hit the ground at the same time since they have the same acceleration due to gravity. I'm not sure but I think they have the same displacement and the ball that was thrown up has a greater distance travelled.

Can someone please explain each of these questions?

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#### BAnders1

Where are the balls thrown from? Using conservation of energy will be much easier than using the kinematic equations (if you aren't already using it).

#### Cyosis

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I think they would hit the ground at the same time since they have the same acceleration due to gravity.
If you stand on a roof and throw one ball down and the other ball up at the same time, would you expect both balls to hit the ground at the same time?

Lets say you throw a ball up with a speed of v0. It goes up and up until it reaches a maximum where v=0. It then begins to fall down again until it reaches the point where you threw it in the air. What is the balls speed at that point?

Where are the balls thrown from? Using conservation of energy will be much easier than using the kinematic equations (if you aren't already using it).
They are both thrown from the same height

#### RoyalCat

The balls both have the same acceleration, but their initial velocity is different. One has a velocity of +v, and the other, a velocity of -v.

Look at the kinematic equations for both of them, and you'll see that they move in different ways.

As for their velocities upon impact, conservation of energy is a good tool for solving that question.
Draw an illustration of their initial conditions, and their final conditions (The point they were thrown at, and the point of impact with the floor) and look at the expressions for their energies.
Here's a hint: They both have the same initial energy, ½mv², since (-v)²=(+v)²

#### afternine

You can answer all of these problems conceptually.

Think about a simple projectile motion trajectory and consider the initial height to be at the ground. When the projectile hits the ground, how does its speed at the instant of impact compare to its initial speed?

What is the definition of displacement? What is the definition of distance? How do they compare?

#### modulus

I don't really think there is any need to get into all of the projectile motion stuff, you can simply do it by using the equations of motion only.
To figure out which one falls with a greater final velocity, you could simply use the equations of motion. Use v^2 = u^2 - 2gs (we'll take gravity as negative, and, 'v' as zero, because, the ball moves upward), and figure out the distance traveled by the ball in the up direction. After that, use your result, in the same equation (v^2 = u^2 - 2gs) for finding the final velocity in the downward direction (this time, take 'u' as zero).
Once you find the final velocity of the ball that was thrown upward, find the final velocity of the ball that was thrown downward. Both the final velocity will turn out to be the same: sqrt(u^2 + 2gh).
Obviously, both will have an equal displacement, as, both started at the height of the roof of the building and ended up on the ground.
Obviously, the ball thrown upwards will take more time, because it traveled more distance (first it move up, then it traveled on the same path as the other ball).