Question

A projectile is projected from a moving truck. Its range depends on:

• A. Truck velocity only
• B. Projectile velocity with respect to truck only
• C. Projectile velocity with respect to ground
• D. Projectile mass

Hint:

In relative motion problems involving trajectories, always transform all velocity components into the frame where the final landing position is being measured (usually the ground frame) before applying standard kinematic formulas.

Answer 1

The Correct Option is C

Concept: Projectile motion is governed by independent horizontal and vertical motions under the influence of uniform gravitational acceleration $g$ acting downwards. The range $R$ of a projectile relative to the fixed ground surface depends fundamentally on its flight path coordinates evaluated relative to the ground reference frame. The standard kinematic equations for range relative to the ground are derived from:

• Time of flight: $T = \frac{2v_{y\text{, ground}}}{g}$

• Horizontal range: $R = v_{x\text{, ground}} \cdot T = \frac{2 \cdot v_{x\text{, ground}} \cdot v_{y\text{, ground}}}{g} $

Step 1: When a projectile is launched from a platform moving with a constant velocity $\vec{V}_{\text{truck}}$ relative to the ground, the velocity vector of the projectile relative to the ground ($\vec{V}_{\text{projectile, ground}}$) is determined using vector addition principles: $$\vec{V}_{\text{projectile, ground}} = \vec{V}_{\text{projectile, truck}} + \vec{V}_{\text{truck, ground}}$$ Therefore, the net horizontal and vertical initial velocity components that dictate its landing coordinates on the earth depend directly on both the velocity of the truck and the launch parameters from the truck.

Step 2: The trajectory path length and landing position ($R$) on the ground are defined strictly with respect to the stationary ground grid. Consequently, the actual mathematical expression for range requires tracking the exact ground-frame velocity components: $$R = \frac{2 \cdot (v_{x\text{, truck}} + v_{\text{truck}}) \cdot v_{y\text{, truck}}}{g}$$ This compound expression represents the components of the initial velocity measured with respect to the ground. Let us examine the individual choices:

• Truck velocity only: Incorrect, because if the projectile is not launched relative to the truck, it remains on the truck.

• Projectile velocity with respect to truck only: Incorrect, because a stationary observer on the ground sees an additional inertial shift due to the truck's forward movement.

• Projectile velocity with respect to ground: Correct, as this single parameter completely captures both the launch velocity and the vehicle's state of motion through vector combination.

• Projectile mass: Incorrect, as mass cancels out in ideal kinematic flight trajectories when ignoring air resistance.
Thus, the parameters combine to rely wholly on its absolute motion parameters measured relative to the target boundary layer, i.e., the ground.