3. In class we found that if something is launched from the ground with fixed initial speed v,, the horizontal distance it travels before hitting the ground (the "range") is maximized if the launch angle is 45° relative to the horizontal and the object travels a distance Often, though, the projectile is launched from a starting height h above the ground, for example when someone throws a ball. If someone throws an object at a starting height h with speed vo at an angle 45° relative to the horizontal, A) Find the time it spends in the air before hitting the ground in terms of the and g. Note you should get two mathematical solutions. Explain in words which solution is the one you want and what the physical meaning of the other solution is. parameters vo, B) In terms of vo, h, and g what is the horizontal distance it travels before hitting the ground? C) Anytime you derive a formula, especially a complicated one like you found in B), you should try to check that it makes sense in multiple ways. One way you can do this is to look at special cases that you already know the answer to: For the formula you found in B), check that it gives the range if the launch height is 0. If it doesn't, go back to A) and B) and check your work.

3. In class we found that if something is launched from the ground with fixed initial speed v,, the horizontal distance it travels before hitting the ground (the "range") is maximized if the launch angle is 45° relative to the horizontal and the object travels a distance Often, though, the projectile is launched from a starting height h above the ground, for example when someone throws a ball. If someone throws an object at a starting height h with speed vo at an angle 45° relative to the horizontal, A) Find the time it spends in the air before hitting the ground in terms of the and g. Note you should get two mathematical solutions. Explain in words which solution is the one you want and what the physical meaning of the other solution is. parameters vo, B) In terms of vo, h, and g what is the horizontal distance it travels before hitting the ground? C) Anytime you derive a formula, especially a complicated one like you found in B), you should try to check that it makes sense in multiple ways. One way you can do this is to look at special cases that you already know the answer to: For the formula you found in B), check that it gives the range if the launch height is 0. If it doesn't, go back to A) and B) and check your work.