Update project

1 files changed, 13 insertions, 2 deletions

diff --git a/projects/projectile/index.html b/projects/projectile/index.html
index 455b6b6..d56edd3 100644
--- a/projects/projectile/index.html
+++ b/projects/projectile/index.html
@@ -15,7 +15,7 @@
   </div>
   <div class="col-sm">
     <div class="input-group">
-      <div class="input-group-text">\(v_0\)</div>
+      <div class="input-group-text">\(v_{0,x},v_{0,y}\)</div>
       <input type="number" class="form-control" v-model="v0" value="50">
       <div class="input-group-text">\(m.s\)</div>
     </div>
@@ -34,4 +34,15 @@
 <br /><br /><br />
 
 <h3>Projectile Motion</h3>
-\[ \int_{0}^{10} \frac{56}{875} \]
+<p>To determine to position of the projectile we should compute the position vector \(\vec{r}=x(t)\vec{i}+y(t)\vec{i}\).</p>
+<h5>\(x(t)\):</h5>
+<p>We know from Newton second law that \(\sum \vec{F} = m\times \vec{a}_x = m\times a_x(t)\vec{i}\)</p>
+<p>However, the projectile as a constant speed along \(\vec{i}\). Hence, \(a_x(t) = 0 \).</p>
+<p>Thus:</p>
+\[ x(t) = \int_{t_0}^t v_{0,x}dt = v_{0,x}t + C = v_{0,x}t + x_0\]
+<h5>\(y(t)\):</h5>
+<p>We know from Newton second law that \(\sum \vec{F} = m\times \vec{a}_y = m\times a_y(t)\vec{i}\)</p>
+<p>The projectile is under the influence of the gravity that is oriented <em>downwarde</em>. Hence, \(a_y(t) = -g \).</p>
+<p>Thus:</p>
+\[ v_y(t) = \int_{t_0}^t a_{y}(t)dt = -gt+C = -gt + v_{0,y}\]
+\[ y(t) = \int_{t_0}^t v_y(t)dt = -\frac{1}{2}gt^2 + v_{0,y}t+C=-\frac{1}{2}gt^2 + v_{0,y}t+y_0\]