Question

an object is projected straight up with an initial velocity of k ft. If the object returns to the ground at t=8 sec., find the height (in feet) of the object at t=7 sec

Answer 1

The general equation is\[h(t) = -\frac{1}{2}gt^2 + v_0 t + h_0\]where \(g\) is the acceleration due to gravity, \(v_0\) is the initial velocity (\(k\) in this problem), and \(h_0\) is the initial height (presumably \(h_0=0\) as unspecified in this problem).

At any time \(t\) where \(h(t)=0\), the object is at ground level. As we are using ft as the unit of measurement, use \(g = 32.2 \text{ft/s}^2\)

That gives you a quadratic to solve:

\[0 = -\frac{1}{2}(32.2)t^2 + k t\]
Find the value of \(k\) that makes that equation true at \(t = 8\). Then plug it into your initial equation and evaluate at \(t=7\) to find the height of the object at \(t=7\) seconds.

Answer 2

can you please explain like step by step please

Answer 3

Well, I thought I did :-)

How about you tell me where my explanation stops making sense, and I'll go from there?

Answer 4

so this is a project but I honestly don't understand any of this so can you break it don how you got 32 and how you plug it in

Answer 5

Okay. Is this for a math class or a physics class? I'm guessing math...

Answer 6

Math, Algebra 2

Answer 7

Fair enough. Well, the first thing you need to know is that if you ignore air resistance, the path of an object thrown up in the air is a parabola. Do you know the general equation for a parabola?

Answer 8

h(t)=−12 gt 2 +v 0 t+h 0 im guessing it is this correct?

Answer 9

A parabola can be expressed as

\[ax^2+bx+c\]where \(a,b,c\) are all constants

I've used different names for those constants to make more physical sense, but the form is the same.

Answer 10

Now, if the parabola represents something thrown up in the air (without sufficient velocity to escape the pull of gravity), the path will look something like this:



Agreed?

Answer 11

Right!

Answer 12

Thank you for working with us, can you do a few more?

Answer 13

The equation I initially gave is the form that a physicist would use. It has 3 components to determine the position/height of the object:

\[-\frac{1}{2}gt^2\]represents the contribution from gravity. This pulls the object down, and gets larger and larger as time goes on (constant acceleration for a longer time means more movement). If you just had an object dropped from the top of a building with no initial velocity, this term alone would give you position relative to the starting point.

\[v_0 t\]represents the contribution from the initial velocity upward. If you had no gravity or other forces acting on the thrown object, this term would give you constant motion with no acceleration. distance = speed * time

\[h_0\]represents your starting height, and in our case is \(0\)

If you add the three together, you get the total "picture" of the motion.

Answer 14

The 32.2 ft/s^2 comes from assuming this problem takes place on the Earth's surface. That's the approximate value of the acceleration due to gravity here.

Answer 15

That's how I came up with the equation. Now we need to do the work!

\[h(t) = -\frac{1}{2}(32.2 \text{ft/s}^2)t^2 + k t + 0\]
We know that at \(t=0\), \(h(t)=0\) because we haven't thrown the ball yet
\[0 = -\frac{1}{2}(32.2 \text{ ft/s}^2)(0)^2 + k(0) + 0\]
but that is all \(0\) for each term, so that didn't help us at all.

Answer 16

We also know that the object is back on the ground at \(t = 8 \text{ s}\), so

\[h(8\text{ s}) = 0 = -\frac{1}{2}(32.2\text{ ft/s}^2)(8\text{ s})^ + k(8\text{ s}) + 0\]
or if we remove the units for clarity:
\[h(8) = 0 = -\frac{1}{2}(32.2)(8)^2 + 8k\]

Can you solve that last equation for \(k\)?