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# Application of derivatives to motion of a projectile

Let the height of a projectile above the ground be given by where is the initial velocity in ft/sec.

1. Show that the average velocity on the interval (with ) is and that the velocity at time is 2. Find the time at which velocity equals zero.
3. Find the velocity of the projectile when it gets back to earth (i.e., when f(t) = 0 again).
4. Find the initial velocity such that the projectile returns to earth in 1 second, 10 seconds, and in seconds.
5. Prove that the projectile undergoes constant acceleration.
6. Find another formula for the height such that the acceleration is a constant -20 ft/sec^2.

1. The average velocity from time to time is given by The instantaneous velocity is the limit of this as or, 2. Let . This implies seconds.
3. We look for the solutions of . The solution corresponds to the time the projectile was fired. So, the projectile returns to earth at time , so evaluating at this time we have, 4. If the projectiles returns in 1 second then we have If the projectile returns in 10 seconds then we have If the projectile returns in seconds then we have (with ), 5. Acceleration is given by . This is a constant.
6. Let . Then and , as requested.