someone please help! I really do not know how to solve this
Please someone help yikes!
Assuming there is no resistance and knowing that a(t)= -9.8 m/s^2
Determine the maximum height and range of a projectile fired at a height of 1.5 meters above the ground with an initial velocity of a 100 meters per second and at an angle of 30 degrees above the horizontal.
Answers in back of book:
maximum height: 129.1 meters
range: 886.3 meters
(how do you go about solving this problem using Vector-Valued Functions such as r(t)= x(t)i + y(t)j etc. )
The equation of motion can be expressed parametrically as you state with
and
(You can derive these from knowing that the acceleration components in both x and y directions are constant. Then , etc.)
We need suitable initial conditions, of course. First off . Also the point acts as an origin for the problem. You can select any one that you like; for my purposes I'm going to choose . That way none of my y values are negative. We need to recognize the and as initial velocity components, so and .
So the parametric equations of motion are:
and
We want the maximum height for the first problem. What is the condition for a moving object to be at an extremum? When the first derivative is 0, of course. So when the particle is at its maximum height. As per my usual, I'll leave it to you to prove that this is indeed a maximum. (Remember you want y(t) not just t.)
The second problem asks about the range. This is the x value (actually value) for when the projectile is at ground level, that is to say y = 0 in my coordinate system. So you need to solve the system of equations:
and
for x.
-Dan