Question

A plane flying horizontally at a height of 1500 m with a velocity of 200 ms⁻¹ passes directly overhead an antiaircraft gun. The angle with the horizontal at which the gun should be fired so that the shell with a muzzle velocity of 400 ms⁻¹ hits the plane, is:

• A. 90°
• B. 60°
• C. 30°
• D. 45°

Answer 1

Answer: (B)

60°

Answer 2

Let the height of the plane be $h = 1500$ m and its horizontal velocity be $v_p = 200$ m/s. The muzzle velocity of the shell is $u_s = 400$ m/s. Let the gun be fired at an angle $\theta$ with the horizontal.

The position of the plane at time $t$ is given by: $x_p(t) = v_p t$ $y_p(t) = h$

The initial velocity components of the shell are: $u_x = u_s \cos \theta$ $u_y = u_s \sin \theta$

The position of the shell at time $t$ is given by: $x_s(t) = u_s \cos \theta \cdot t$ $y_s(t) = u_s \sin \theta \cdot t - \frac{1}{2} g t^2$

For the shell to hit the plane, their positions must be the same at some time $t > 0$. Equating the horizontal positions: $x_p(t) = x_s(t)$ $v_p t = u_s \cos \theta \cdot t$

Since $t > 0$, we can divide by $t$: $v_p = u_s \cos \theta$

This equation gives the condition for the horizontal velocities to match, which is necessary for the shell to intercept the plane. From this, we can find the angle $\theta$: $\cos \theta = \frac{v_p}{u_s}$ $\cos \theta = \frac{200 \text{ ms}^{-1}}{400 \text{ ms}^{-1}}$ $\cos \theta = \frac{1}{2}$

Therefore, the angle $\theta$ is: $\theta = \cos^{-1}\left(\frac{1}{2}\right) = 60^\circ$

For the interception to be possible, the shell must also reach the height $h$ at some time $t$. This requires the discriminant of the quadratic equation for $t$ derived from the vertical motion to be non-negative. The condition is $u_s^2 \sin^2 \theta - 2gh \ge 0$, or equivalently $u_s^2 - v_p^2 \ge 2gh$. Using $g \approx 10$ m/s², $h = 1500$ m, $v_p = 200$ m/s, $u_s = 400$ m/s: $u_s^2 - v_p^2 = 400^2 - 200^2 = 160000 - 40000 = 120000$ $2gh = 2 \times 10 \times 1500 = 30000$ Since $120000 \ge 30000$, the condition is satisfied, and interception is possible.

The angle with the horizontal at which the gun should be fired is $60^\circ$.