Question

a projectile is fired from ground level with an initial speed of 100 m/sec and an angle of elevation of 30 degrees. use that the acceleration due to gravity is 9.8 m/sec². the range of the projectile is meters. the maximum height of the projectile is meters. the speed with which the projectile hits the ground is m/sec.

Explanation:

Step1: Find initial vertical and horizontal velocities

The initial speed $v_0 = 100$ m/s and the angle of elevation $\theta=30^{\circ}$. The initial vertical velocity $v_{0y}=v_0\sin\theta$ and the initial horizontal velocity $v_{0x}=v_0\cos\theta$. So $v_{0y}=100\sin30^{\circ}= 50$ m/s and $v_{0x}=100\cos30^{\circ}=50\sqrt{3}$ m/s.

Step2: Calculate the maximum height

Use the kinematic - equation $v_y^2 = v_{0y}^2-2gh$. At the maximum - height, $v_y = 0$. Rearranging for $h$ gives $h=\frac{v_{0y}^2}{2g}$. Substituting $v_{0y}=50$ m/s and $g = 9.8$ m/s², we have $h=\frac{50^2}{2\times9.8}=\frac{2500}{19.6}\approx127.55$ m.

Step3: Calculate the time of flight

Use the kinematic - equation $v_y=v_{0y}-gt$. When the projectile hits the ground, the vertical displacement $y - y_0 = 0$. Using $y - y_0=v_{0y}t-\frac{1}{2}gt^2$, factoring out $t$ gives $t(v_{0y}-\frac{1}{2}gt)=0$. One solution is $t = 0$ (corresponds to the launch time), and the other is $t=\frac{2v_{0y}}{g}$. Substituting $v_{0y}=50$ m/s and $g = 9.8$ m/s², we get $t=\frac{2\times50}{9.8}\approx10.2$ s.

Step4: Calculate the range

The range $R$ is given by the horizontal motion, and since there is no acceleration in the horizontal direction ($a_x = 0$), $R = v_{0x}t$. Substituting $v_{0x}=50\sqrt{3}$ m/s and $t\approx10.2$ s, we have $R = 50\sqrt{3}\times10.2\approx883.3$ m.

Step5: Analyze the speed when hitting the ground

By the conservation of mechanical energy, the speed of the projectile when it hits the ground is the same as the initial speed because the initial and final heights are the same. So the speed when hitting the ground is $v = 100$ m/s.

Answer:

The maximum height of the projectile is approximately $127.55$ meters.
The range of the projectile is approximately $883.3$ meters.
The speed with which the projectile hits the ground is $100$ m/s.