Height equation

Question

A clown is standing on a cliff that is 528 ft tall. The clown hurls a tomato into the air with a velocity of 128ft/sec. How fast is the tomato traveling when it hits the ground below?

 

I know the equation is h=-16t^2+128t+528 because the teacher told us, but i don't get where -16 came from as it is not a number listed in the word problem.

Answer 1

Have you all started integration yet, in your calculus class?

 

It comes from the acceleration due to gravity.   Gravity pulls things back down to the ground at an acceleration of approximately 32 ft/s².  Because it is pulling DOWN, it is negative.  When you integrate acceleration to find velocity, you get that the velocity is -32t + initial velocity.

 

When you integrate the velocity to get the height, you get -32(t²/2) + 128t + initial height.

 

This simplifies to  -16t² + 128t +528.

 

The height of the ground is 0, so solve the equation  0 = -16t²+128t+528

 

You will get two answers, only one will make sense.

 

I get that the time for the height to be 0 is at t=-3 and t = 11.  Since we rarely travel backwards in time, I would say that the tomato hits the ground when t = 11 seconds.

 

But that was not the question.

 

The question was how fast was it traveling.  So plug the t we found into the velocity equation (derivative of h = dh/dt = -32t+128 = velocity equation) to find the velocity it was traveling when it hit the ground.  

 

I get (-32)(11)+128 = -224 ft/sec

 

Why is it negative?  What direction was it traveling when it hit the ground?   DOWN, so that is why it is negative.