Why do all three balls land with the same Kinetic energy?

[csx]

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3 balls are shot from a cliff 100 meters high. Ball I is shot horizontally. Ball II at 30 degrees and ball III at 60 degrees. All are shot at same initial speed of 75 m/s.

I get that they all start with the same Potential energy (100 meters up on cliff) and if they are all shot at a speed of 75 m/s....shouldn't the balls shot at an angle have more kinetic energy upon landing as they gain potential energy being shot up.

Answer says they land all with the same KE.

 

[BenzylAcetate]

Velocity has both a vertical and horizontal component. The angled balls will have greater vertical velocity but smaller horizontal velocity.

Think of it this way, what are all the forces adding energy to the balls? There is the initial potential energy from being at a height of 100m and energy imparted by the explosive force of the cannon. If all three balls have the same speed out of the cannon, then all three balls receive the same amount of energy from the cannon. There are no other forces adding or removing energy after that point, just gravity converting potential energy to kinetic.

 

[JMMTB]

Like you said they all start with the same PE and they all start with the same KE (velocity is the same for all) so they have to end with the same KE. Energy is a scalar with no direction is it is completely independent of the angle they are shot from.

You are correct that as they go up they gain PE, but at the same time there is an equal loss of KE. Once the ball has returned back to the height it was fired from that "gain" in PE has been converted back into the same initial amount of KE thus netting no overall gain in total energy just for going up.