Question

Which of the following graphs represents the speed $v$ of a projectile as a function of time $t$

• A. Graph (B)
• B. Graph (C)
• C. Graph (D)
• D. Graph (E)
• E.

Hint:

Remember that in projectile motion (without air resistance), the speed is never zero at any point in the flight because the horizontal velocity remains constant.

Answer 1

Answer: (E)

Concept:
In projectile motion, the velocity components are: [itemsep=8pt]
• Horizontal velocity ($v_x$): $u \cos \theta$ (Constant throughout the motion).
• Vertical velocity ($v_y$): $u \sin \theta - gt$ (Changes linearly with time).
• Speed ($v$): Resultant magnitude, $v = \sqrt{v_x^2 + v_y^2}$.

Step 1: Analyze the speed equation.
Substituting the components: \[ v = \sqrt{(u \cos \theta)^2 + (u \sin \theta - gt)^2} \] At $t=0$, $v = u$, which is a non-zero positive value. As $t$ increases, the vertical component $v_y$ decreases until it becomes zero at the highest point ($t = u \sin \theta / g$). At this point, the speed is minimum ($v = u \cos \theta$) but not zero.

Step 2: Curve analysis.
After the highest point, $v_y$ becomes negative and increases in magnitude, so speed increases again in a curved (non-linear) manner starting from a non-zero value. Hence Graph (E) matches.