A rock is thrown up through the air at an angle. Which of the following quantities is equal in size to its minimum speed?

Question

BlackTom AI · Accepted Answer

At the top of the projectile's path the vertical component of velocity is zero, so the instantaneous speed equals the horizontal component alone.

Relevant concepts:
- Projectile motion can be split into independent horizontal and vertical components.
- Horizontal velocity (vx) is constant when air resistance is negligible.
- Vertical velocity (vy) changes due to gravity and becomes zero at the highest point.
- Speed is the magnitude of the velocity vector: v = sqrt(vx^2 + vy^2).

Step-by-step reasoning and calculation:
1. Let the initial speed be v0 at launch angle θ. Resolve into components:
   - v0x = v0 cos θ (horizontal component)
   - v0y = v0 sin θ (vertical component)
2. Because no horizontal acceleration, vx remains v0 cos θ throughout the flight.
3. At the trajectory’s highest point vy = 0. The speed there is
   v_top = sqrt(vx^2 + vy^2) = sqrt((v0 cos θ)^2 + 0^2) = v0 cos θ.
4. Since vy is nonzero everywhere except at the top, the speed elsewhere is sqrt((v0 cos θ)^2 + (vy)^2) ≥ v0 cos θ. Thus v_top is the minimum speed.

Why the correct answer is correct:
- The minimum speed occurs at the top and equals v0 cos θ, which is the horizontal component of the initial velocity. Because horizontal component is constant, it is also the horizontal component just before hitting the ground — so option (a) matches the minimum speed.

Why other options are incorrect:
- (b) The initial speed v0 is larger than v0 cos θ unless θ = 0°.
- (c) Speed is not zero because horizontal component remains nonzero.
- (d) The initial vertical component v0 sin θ is generally different (and typically larger for upward launches).
- (e) The final speed just before hitting the ground equals the initial speed in symmetric launch/landing cases, not the minimum.

Conclusion: The minimum speed equals the horizontal component of the initial velocity, so option (a) is correct.

A rock is thrown up through the air at an angle. Which of the following quantities is equal in size to its minimum speed?Single choice

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