Question

a projectile is fired with an initial speed of 210 m/s and angle of elevation 60°. the projectile is fired from a position 80 m above the ground. (use
(a) find the range (in m) of the projectile.

m

(b) find the maximum height (in m) reached.

m

(c) find the speed (in m/s) at impact.

m/s

Explanation:

Step1: Resolve initial velocity

The initial velocity \(v_0 = 210\ m/s\), angle \(\theta=60^{\circ}\).
The horizontal component \(v_{0x}=v_0\cos\theta = 210\cos60^{\circ}=105\ m/s\).
The vertical component \(v_{0y}=v_0\sin\theta = 210\sin60^{\circ}=105\sqrt{3}\ m/s\).

Step2: Use vertical motion equation for time of flight

The vertical displacement \(y - y_0=- 80\ m\) (taking ground as \(y = 0\), \(y_0 = 80\ m\)), \(a=-g=-9.8\ m/s^{2}\).
Using \(y - y_0=v_{0y}t+\frac{1}{2}at^{2}\), we have \(-80 = 105\sqrt{3}t-4.9t^{2}\).
Solving the quadratic equation \(4.9t^{2}-105\sqrt{3}t - 80=0\).
Using the quadratic formula \(t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) where \(a = 4.9\), \(b=-105\sqrt{3}\), \(c = - 80\).
\(t=\frac{105\sqrt{3}\pm\sqrt{(105\sqrt{3})^{2}-4\times4.9\times(-80)}}{2\times4.9}\).
We take the positive root \(t\approx38.0\ s\).

Step3: Calculate range (a)

Range \(R = v_{0x}t\).
Substituting \(v_{0x}=105\ m/s\) and \(t\approx38.0\ s\), \(R=105\times38.0 = 3990\ m\).

Step4: Calculate maximum height (b)

At maximum height, \(v_y = 0\). Using \(v_y^{2}-v_{0y}^{2}=2a(y - y_0)\).
\(0-(105\sqrt{3})^{2}=2\times(-9.8)(H - 80)\).
\(H-80=\frac{(105\sqrt{3})^{2}}{2\times9.8}\).
\(H=\frac{(105\sqrt{3})^{2}}{2\times9.8}+80\).
\(H=\frac{105^{2}\times3}{19.6}+80=\frac{33075}{19.6}+80\approx1700 + 80=1780\ m\).

Step5: Calculate speed at impact (c)

Horizontal component \(v_x=v_{0x}=105\ m/s\).
Using \(v_y^{2}-v_{0y}^{2}=2a(y - y_0)\) with \(y - y_0=-80\ m\).
\(v_y^{2}=(105\sqrt{3})^{2}+2\times(-9.8)\times(-80)\).
\(v_y^{2}=33075 + 1568=34643\).
\(v_y=\sqrt{34643}\approx186.1\ m/s\).
Speed \(v=\sqrt{v_x^{2}+v_y^{2}}=\sqrt{105^{2}+34643}=\sqrt{11025 + 34643}=\sqrt{45668}\approx214\ m/s\).

Answer:

(a) \(3990\ m\)
(b) \(1780\ m\)
(c) \(214\ m/s\)