Projectile Motion and Integration

[auk411]

Homework Statement

A projectile is launched from the origin a distance h above the Earth's surface with an initial velocity having speed v_0 and direction Theta_0 with respect to horizontal x-axis. Obtain an integral expression for the distance s the projectile travels along its path until it hits the ground at y = -h. Evaluate all derivatives in the integrand but do not evaluate the integral. Express any non-givens in terms of the givens of the stated problem.

Homework Equations

y-y_0 = V_0cos(theta)(t) - .5gt^2
x-x_0 = V_osin(theta)(t) - .5gt^2

The Attempt at a Solution

I do not know where to start. I have solved projectile motion problems before, but not like this. I'm thinking I might need to use a line integral. But other than that, I really don't know how to even begin. Help?

 

[Tomer]

Erm, either I'm missing something, or there's no integral involved.
The only way I could see an integral popping is by assuming that h is very big, which would mean that "g" as we know it isn't constant.
Otherwise you could solve it with two simple motion equations. Notice that there are two mistakes in the equations you've written.
1. You got the components of the velocity messed up - it should be sin for y, and cos for x. Draw it and make sure you know why.
2. There is no acceleration on the x-axis - only on the y axis. Therefore the second equation is wrong and doesn't fit. Which does?

So, as I've said, solving it "normally" wouldn't give rise to any integrals (but to an ugly quadratic equation), but assuming h is big enough would force you to use Newton's second law, and that's another story... ! :-)

 

[ehild]

A projectile is launched from the origin a distance h above the Earth's surface with an initial velocity having speed v_0 and direction Theta_0 with respect to horizontal x-axis. Obtain an integral expression for the distance s the projectile travels along its path until it hits the ground at y = -h. Evaluate all derivatives in the integrand but do not evaluate the integral.

The problem asks the distance traveled along the path. It is not the same as the displacement.

The distance traveled is the length of arc between the initial and final positions. It is a line integral, S=∫ds. You can express ds, the line element, with the time or with x. What have you learned about line integrals?

ehild

 

[Tomer]

Sorry - have missed that. Good thing I've volunteered the option: "Either I'm missing something"... :-)