True or false about velocity and acceleration

Problem

True or false?

• (a) A body can have zero velocity and non-zero acceleration at a certain instant.
• (b) If the acceleration is constant, the velocity must be constant.
• (c) A body moving with uniform rectilinear motion has an average velocity different from its instantaneous velocity.
• (d) In the $v$–$t$ graph, the area under the curve represents the distance travelled.

Solution

(a) True. Velocity and acceleration are independent quantities at the same instant: acceleration tells us how the velocity is changing, not what it is worth. The classic example is a stone thrown upward: at the highest point of the trajectory $v = 0$, but the acceleration is still $g \approx 9{.}81\;\text{m/s}^2$ downward, and it is precisely this that makes the stone start falling back down.

(b) False. Constant acceleration means the velocity changes at a constant rate, not that it stays fixed. If $a$ is constant and non-zero, the velocity varies linearly with time, $v(t) = v_0 + a t$. The velocity is constant only in the special case $a = 0$.

(c) False. In uniform rectilinear motion the instantaneous velocity is the same at every instant; consequently its average over any interval coincides with that value. Average and instantaneous velocity are equal. (This is the only case in which this always holds.)

(d) True. In a velocity-time graph, the area between the curve and the time axis is numerically equal to the distance travelled: for a leg at constant velocity the area is the rectangle $v \cdot \Delta t$, and in general summing infinitesimal areas $v\,\mathrm{d}t$ gives the total distance.

$\ev{\text{(a) T} \quad \text{(b) F} \quad \text{(c) F} \quad \text{(d) T}}$

Collegamenti

Argomenti: Kinematics Concetti: Acceleration · Free fall