Question

A projectile is projected with a velocity of \( 20 \, {ms}^{-1} \) at an angle of 45° to the horizontal. After some time its velocity vector makes an angle of 30° to the horizontal. Its speed at this instant (in \( {ms}^{-1} \)) is:

• A. \( 10\sqrt{\frac{2}{3}} \)
• B. \( \frac{20}{\sqrt{3}} \)
• C. \( 20\sqrt{\frac{2}{3}} \)
• D. \( 10\sqrt{2} \)
• E. \( 10\sqrt{3} \)

Hint:

In projectile motion, use vector components to resolve velocity at any point and find the resultant speed.

Answer 1

The Correct Option is C

For projectile motion, the velocity at any point can be broken into two components: horizontal and vertical. The initial velocity \( u \) is given as \( 20 \, {m/s} \) at an angle of 45° to the horizontal. 
Step 1: Resolve the velocity into horizontal and vertical components. 
- The horizontal component of the velocity is: \[ u_x = u \cos \theta = 20 \cos 45^\circ = 20 \times \frac{1}{\sqrt{2}} = 10\sqrt{2} \, {m/s}. \] - The vertical component of the velocity is: \[ u_y = u \sin \theta = 20 \sin 45^\circ = 20 \times \frac{1}{\sqrt{2}} = 10\sqrt{2} \, {m/s}. \] 
Step 2: At the given point, the angle of the velocity vector is 30° to the horizontal. The velocity components at this point are: 
- The horizontal velocity remains the same, as horizontal velocity does not change in projectile motion: \[ v_x = u_x = 10\sqrt{2} \, {m/s}. \] 
- The vertical velocity component is given by: \[ v_y = v \sin 30^\circ, \] where \( v \) is the speed at this instant. We will solve for \( v \) using the equation for vertical velocity in projectile motion. Since vertical velocity changes due to gravity, we can use the fact that the total vertical velocity at this point is given by: \[ v_y = u_y - g t. \] At the moment when the angle is 30°, the speed \( v \) can be determined using the following relation for projectile motion: \[ v = \sqrt{v_x^2 + v_y^2} = \sqrt{(10\sqrt{2})^2 + \left(10 \times \frac{\sqrt{2}}{3}\right)^2}. \] 
Step 3: Simplify: \[ v = \sqrt{(10\sqrt{2})^2 + \left(10 \times \frac{\sqrt{2}}{3}\right)^2} = \sqrt{200 + 66.67} = \sqrt{266.67} \approx 20\sqrt{\frac{2}{3}}. \]