How Do You Solve a Projectile Motion Problem with Calculus?

[dominus96]

Homework Statement

A ball is tossed up in the air from the top of a tower. The function for this is
y = -16t² + 64t + 80. The problems are:

a) What is the velocity of the ball after 4 seconds?
b) What was the initial velocity that the ball was tossed?
c) When will the ball reach its maximum height?

Homework Equations

v=v₀ + at, s = v₀t + at²/2
And the derivative equation CNX^N-1, dx/dy

The Attempt at a Solution

Well, I calculated the velocity at 1 second (a different part of the problem) and got 32 m/s. 32 is also the acceleration I got from the 2nd derivative, which doesn't make sense since shouldn't it be -9.81?

I know you use the 1st deriv to find instant velocity, but how do you figure out the initial velocity? do you just use those old formulas that I listed, or do you apply calculus to this? I know how to find instant velocity using derivative, but don't know how to do velocity after a period of time (4 seconds). An part c) I have no idea.

 

[alphysicist]

Homework Statement

A ball is tossed up in the air from the top of a tower. The function for this is
y = -16t² + 64t + 80. The problems are:

a) What is the velocity of the ball after 4 seconds?
b) What was the initial velocity that the ball was tossed?
c) When will the ball reach its maximum height?

Homework Equations

v=v₀ + at, s = v₀t + at²/2
And the derivative equation CNX^N-1, dx/dy

The Attempt at a Solution

Well, I calculated the velocity at 1 second (a different part of the problem) and got 32 m/s. 32 is also the acceleration I got from the 2nd derivative, which doesn't make sense since shouldn't it be -9.81?

That equation is not in metric units; the acceleration due to gravity is (about) 32 ft/s^2 downwards.

I know you use the 1st deriv to find instant velocity, but how do you figure out the initial velocity?

In the first equation (the one in the problem itself), what are the different terms of the equation? What is the general equation that that one comes from?

do you just use those old formulas that I listed, or do you apply calculus to this? I know how to find instant velocity using derivative, but don't know how to do velocity after a period of time (4 seconds). An part c) I have no idea.

You can find the answer using just these equations and the ideas of free fall.

 

[dominus96]

Alright I got 96 as the initial velocity, and 0 as the velocity after 4 seconds. Are those right?

 

[alphysicist]

Alright I got 96 as the initial velocity, and 0 as the velocity after 4 seconds. Are those right?

No, I don't believe those answers are correct. What did you do to find them? Remember to compare the equation in the problem with the corresponding one in your "relevant equations" section. What do $s$, $v_o$, and $a$ mean?