Velocity–time graphs Transitional Reasoning Practice

Questions

1. 1.

   A motorized lift platform moves vertically along a straight track inside a test facility. Its velocity–time log reveals three successive stages over a 16 s interval: it accelerates from rest at t = 0 s to +8 m/s at t = 4 s, maintains a constant velocity of +8 m/s until t = 10 s, and then undergoes a constant acceleration until its velocity reaches -4 m/s at t = 16 s. What is the platform's displacement during the period in which its speed is decreasing, and what is its net displacement over the entire 16 s test?

   • A) +12 m and +84 m
   • B) +12 m and +76 m
   • C) +16 m and +76 m
   • D) +16 m and +84 m
2. 2.

   An automated research carriage moves along a straight calibrated track. Its velocity–time graph shows that from $t = 0\text{ s}$ to $t = 6\text{ s}$, its velocity changes linearly from $-12\text{ m/s}$ to $+6\text{ m/s}$. From $t = 6\text{ s}$ to $t = 10\text{ s}$, it continues with a different constant acceleration such that its displacement over this four-second interval is $+40\text{ m}$. What are the carriage's acceleration during the first interval and its final velocity at $t = 10\text{ s}$?

   • A) Acceleration: $+1\text{ m/s}^2$; Final velocity: $+10\text{ m/s}$
   • B) Acceleration: $+3\text{ m/s}^2$; Final velocity: $+14\text{ m/s}$
   • C) Acceleration: $+3\text{ m/s}^2$; Final velocity: $+8\text{ m/s}$
   • D) Acceleration: $-1\text{ m/s}^2$; Final velocity: $+14\text{ m/s}$
3. 3.

   An automated rail inspection cart begins from rest at the origin (x = 0 m) at t = 0 s and travels along a straight track. Its telemetry log records that it accelerates uniformly to +10 m/s at t = 5 s, decelerates uniformly to come momentarily to rest at t = 8 s, accelerates in the reverse direction to -4 m/s at t = 12 s, and finally decelerates uniformly until stopping at t = 14 s. At what time does the cart reach its maximum distance in the positive direction from the origin, and what is its position at that instant?

   • A) t = 5 s and x = +25 m
   • B) t = 8 s and x = +15 m
   • C) t = 14 s and x = +28 m
   • D) t = 8 s and x = +40 m
4. 4.

   A mobile robotic platform starts from rest at the origin ($x = 0\text{ m}$) at $t = 0\text{ s}$ and travels along a straight track. Its velocity–time log shows that it accelerates uniformly to $+8\text{ m/s}$ at $t = 2\text{ s}$, maintains this constant velocity until $t = 4\text{ s}$, and then decelerates uniformly to come momentarily to rest at $t = 6\text{ s}$. Immediately after, it accelerates uniformly in the reverse direction to $-4\text{ m/s}$ at $t = 8\text{ s}$ and continues at this constant velocity of $-4\text{ m/s}$ thereafter. At what time $t$ does the platform return to its starting position, and what total distance has it traveled at that moment?

   • A) $t = 12\text{ s}$ and total distance = $48\text{ m}$
   • B) $t = 14\text{ s}$ and total distance = $56\text{ m}$
   • C) $t = 15\text{ s}$ and total distance = $64\text{ m}$
   • D) $t = 16\text{ s}$ and total distance = $68\text{ m}$
5. 5.

   A diagnostic sensor records the motion of an automated shuttle along a straight calibration track over a 10 s interval. The shuttle's velocity–time graph is a single straight line that starts at -6 m/s at t = 0 s, crosses 0 m/s at t = 4 s, and reaches +9 m/s at t = 10 s. Based on this velocity–time graph, what are the total distance traveled by the shuttle and its average velocity over the entire 10 s test?

   • A) Total distance = 15 m, Average velocity = +3.9 m/s
   • B) Total distance = 27 m, Average velocity = +2.7 m/s
   • C) Total distance = 15 m, Average velocity = +1.5 m/s
   • D) Total distance = 39 m, Average velocity = +1.5 m/s
6. 6.

   A telemetry sensor on a straight test track logs the velocity of an electric transport cart over a 5 s period. The cart has an initial velocity of 4 m/s at t = 0 s, accelerates uniformly at +2 m/s² for 3 s to reach a peak velocity of 10 m/s, and then immediately decelerates uniformly at a constant rate until coming to rest at t = 5 s. Based on this velocity–time data, what are the total displacement of the cart and its average acceleration over the entire 5 s interval?

   • A) 25 m and -0.8 m/s²
   • B) 31 m and -1.5 m/s²
   • C) 31 m and -0.8 m/s²
   • D) 35 m and +0.8 m/s²
7. 7.

   An autonomous inspection robot travels along a straight track, starting from rest at 0 s and accelerating uniformly to a steady velocity $v$ at 4 s. It maintains this constant velocity until 10 s, after which it decelerates uniformly to come to a complete stop at 12 s. If the total distance covered during this 12 s journey is 72 m, what are the steady velocity $v$ and the magnitude of the deceleration during the final 2 s?

   • A) 6 m/s and 3 m/s²
   • B) 12 m/s and 6 m/s²
   • C) 8 m/s and 2 m/s²
   • D) 8 m/s and 4 m/s²
8. 8.

   Two automated sorting carriers, X and Y, move along parallel straight tracks in a warehouse, both starting from the same baseline at $t = 0\text{ s}$. Carrier X travels at a constant velocity of $6\text{ m/s}$ throughout an $8\text{ s}$ test. Carrier Y starts from rest and accelerates uniformly to $12\text{ m/s}$ at $t = 4\text{ s}$, then continues at that constant velocity until $t = 8\text{ s}$. Based on their velocity–time graphs, at what time does Carrier Y catch up to Carrier X, and what is their displacement from the baseline at that moment?

   • A) At $t = 2\text{ s}$, with a displacement of $12\text{ m}$
   • B) At $t = 4\text{ s}$, with a displacement of $24\text{ m}$
   • C) At $t = 6\text{ s}$, with a displacement of $36\text{ m}$
   • D) At $t = 8\text{ s}$, with a displacement of $48\text{ m}$
9. 9.

   A sensor-guided test cart moves along a straight track, and its motion is tracked over a 12-second interval. Starting from rest at the origin at $t = 0\text{ s}$, the cart accelerates uniformly to a velocity of $+6\text{ m/s}$ at $t = 4\text{ s}$, then uniformly decelerates to $0\text{ m/s}$ at $t = 7\text{ s}$ and reaches $-2\text{ m/s}$ at $t = 8\text{ s}$, after which it maintains a constant velocity of $-2\text{ m/s}$ until $t = 12\text{ s}$. Based on this velocity–time data, which statement correctly describes the cart's motion?

   • A) The cart's acceleration is zero at $t = 7\text{ s}$ because its instantaneous velocity is zero.
   • B) The cart changes its direction of motion at $t = 4\text{ s}$ when its speed begins to decrease.
   • C) The total distance traveled by the cart throughout the 12-second test is equal to $12\text{ m}$.
   • D) The cart is farthest from its starting position at $t = 7\text{ s}$, reaching a displacement of $+21\text{ m}$.

Answers and step-by-step solutions appear after you finish the test.