# Determing Height To Which A Projectile Will Rise

1. Jan 28, 2013

### Bashyboy

1. The problem statement, all variables and given/known data
At the Earth's surface, a projectile is launched straight up at a speed of 9.6 km/s. To what height will it rise? Ignore air resistance and the rotation of the Earth.

2. Relevant equations
Conservation of mechanical energy $K_i + U_i = K_f + U_f$

3. The attempt at a solution
I am quite confident that I chose the correct formula to solve this problem; however, I have a few questions regarding the formula itself.

I know that $K_i$ and $U_i$ won't be zero, because the projectile has an initial velocity and it is above the surface of the earth. Would $K_f$ be zero, though; because we are considering the highest point the projectile will reach, and gravity will necessary reduce its speed to zero, corresponding to the highest point. Because $K_f = 0$, $U_f$ can't equal zero, right? What would the situation be like if it was negative, how about positive?

2. Jan 28, 2013

### PeterO

There seems to be a contradiction in the two pieces I have shaded red.

Energy is a scalar - so is neither positive nor negative. That makes your final sentence rather curious.

3. Jan 28, 2013

### tms

Last time I checked, scalars had signs.

4. Jan 28, 2013

### tms

Only differences between potential energies have physical meaning, so the zero point is arbitrary; you can set the potential energy to zero at any convenient point.
Well, would it?

5. Jan 28, 2013

### Bashyboy

6. Jan 28, 2013

### Bashyboy

Yes, I suppose I can see how the ambiguity could arise. What I intended for the statement, "..it is above the surface of the earth," was that the projectile wasn't below the the surface, that is, underground.

7. Jan 28, 2013

### PeterO

Above the surface and below the surface represent only 2 of the 3 possibilities for the projectile's original position.

The opening words referred to that 3rd possibility.

8. Jan 28, 2013

### tms

Surely you have enough knowledge by now to say what the kinetic energy of an object with zero velocity is.

9. Jan 28, 2013

### HallsofIvy

Staff Emeritus
If you really don't know how to calculate kinetic energy, rather than using energy, you could solve this problem with the basic $h(t)= h_0+ v_0t- (g/2)t^2$.

You can take $h_0$ to be 0 at the surface of the earth and you are given that $v_0= 9.6$. Of course, g= 9.82 m/s^2, approximately.

10. Jan 28, 2013

### PeterO

Given the initial velocity of 9.6 km/s this mass is going to rise more than 4000 km. It may not be reasonable to assume that g has a constant value to that elevation.

You probably need a potential energy formula like U = -GMm/R