Determing Height To Which A Projectile Will Rise

[Bashyboy]

Homework Statement

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.

Homework Equations

Conservation of mechanical energy $K_i + U_i = K_f + U_f$

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?

 

[PeterO]

Homework Statement

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.

Homework Equations

Conservation of mechanical energy $K_i + U_i = K_f + U_f$

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?

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.

 

[tms]

Energy is a scalar - so is neither positive nor negative.

Last time I checked, scalars had signs.

 

[tms]

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.

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.

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.

Well, would it?

 

[Bashyboy]

Well, would it?

If had the knowledge to answer your question, then I wouldn't have asked my question in the first place.

 

[Bashyboy]

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

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.

 

[PeterO]

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.

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.

 

[tms]

If had the knowledge to answer your question, then I wouldn't have asked my question in the first place.

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

 

[HallsofIvy]

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.

 

[PeterO]

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.

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