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Loic Guegan 2021-10-04 17:53:12 +02:00
parent a31149727b
commit 83f94fb969
3 changed files with 23 additions and 12 deletions
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projects/projectile/index.html
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@ -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>
\[ v_y(t) = \int_{t_0}^t a_{y}(t)dt = -gt+C = -gt + v_{0,y}\]
\[ y(t) = \int_{t_0}^t v_y(t)dt = -\frac{1}{2}gt^2 + v_{0,y}t+C=-\frac{1}{2}gt^2 + v_{0,y}t+y_0\]