Instantaneous Velocity and Instantaneous Speed (dx/dt)
Physics · Motion In A Straight Line · NEET
Instantaneous velocity is the velocity of an object at one exact instant of time. It is defined as the limit of average velocity as the time interval shrinks to zero, so v = dx/dt (the slope of the position-time graph at that point). Instantaneous speed is simply the magnitude of instantaneous velocity, so speed = |v|. Memory hook: "instant = slope right now" — average uses a whole interval, instantaneous uses a single point.
The instantaneous velocity at instant t is the slope of the tangent line to the position-time graph at point P; as the interval shrinks to zero the chord becomes this tangent, giving v = dx/dt.
Your doubts, answered
Is instantaneous speed always equal to the magnitude of instantaneous velocity?
Yes. This is a key NCERT point and a favourite NEET trap. For AVERAGE quantities, average speed can be greater than the magnitude of average velocity (because distance is greater than or equal to displacement). But at a single instant there is no interval for the path to bend, so instantaneous speed = |instantaneous velocity| exactly. Example: if v = -24 m/s, the speed is 24 m/s.
How do I find instantaneous velocity from a position-time equation like x = 3t^2 + 2t?
Differentiate x with respect to t. Here v = dx/dt = 6t + 2. To get the velocity at a specific instant, plug that value of t in. At t = 2 s, v = 6(2) + 2 = 14 m/s. This differentiation step is the whole idea of instantaneous velocity, and it is tested directly in NEET (see the 2016 PYQ below).
Why is instantaneous velocity the slope of the position-time (x-t) graph?
Average velocity between two points is the slope of the line joining them (rise over run = change in x over change in t). As the time interval delta t shrinks to zero, that chord becomes the tangent line at one point. The slope of that tangent is dx/dt, which is exactly the instantaneous velocity at that instant.
Can instantaneous velocity be negative but instantaneous speed still positive?
Yes. Velocity is a vector, so its sign shows direction (a negative value means motion in the negative x-direction). Speed is the magnitude, so it can never be negative. A particle with v = -5 m/s is moving backward at a speed of 5 m/s.
When is instantaneous velocity equal to average velocity?
Only for uniform motion (constant velocity). NCERT states this directly: for uniform motion the velocity is the same as the average velocity at all instants. For any non-uniform motion, the instantaneous value keeps changing, so it usually differs from the average over an interval.
⚠️ The NEET trap ✗ Instantaneous speed can be greater than the magnitude of instantaneous velocity, just like average speed exceeds average velocity. ✓ At a single instant they are always equal: instantaneous speed = |instantaneous velocity|. The 'speed can be greater' rule applies only to AVERAGE quantities over an interval. 🧠 Interval bends the path, an instant does not. Average speed vs velocity can differ; instantaneous speed and |velocity| never differ.
Real NEET questions
2016
Two cars P and Q start from a point at the same time in a straight line and their positions are represented by x_P = at + bt^2 and x_Q = ft - t^2. At what time do the cars have the same velocity?
A · (f - a) / [2(1 + b)] ✓
B · (a + f) / [2(b - 1)]
C · (a - f) / (1 + b)
D · (a + f) / [2(1 + b)]
Solution: Instantaneous velocity is v = dx/dt, so differentiate each position. v_P = dx_P/dt = a + 2bt and v_Q = dx_Q/dt = f - 2t. Set them equal for the same velocity: a + 2bt = f - 2t. Group the t terms: 2bt + 2t = f - a, so t(2b + 2) = f - a. Therefore t = (f - a) / [2(1 + b)]. Answer: A.
2026
A particle moves along a straight line with position s(t) = alpha*t^2 - beta*t + gamma, where alpha = 1 m/s^2, beta = 6 m/s, gamma = 5 m. The average speed of the particle from t = 0 to t = 6 s is:
A · 12 m/s
B · 6 m/s
C · 3 m/s ✓
D · 0 m/s
Solution: First find instantaneous velocity: v = ds/dt = 2t - 6. It is zero at t = 3 s, so the particle reverses direction there (velocity changes sign). Because it turns back, distance is NOT the same as displacement. Distance from 0 to 3 s and from 3 to 6 s, using the v-t triangle areas: d = (1/2)(3)(6) + (1/2)(3)(6) = 9 + 9 = 18 m. Average speed = distance / time = 18 / 6 = 3 m/s. Answer: C. Trap: average velocity here is 0 (option D), because s(0) = s(6).
Solved Motion In A Straight Line NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
v = lim (delta t goes to 0) of (delta x / delta t) = dx/dt. In words, it is the rate of change of position with time at a single instant, found by differentiating the position function.
What is the difference between instantaneous speed and instantaneous velocity?
Instantaneous velocity is a vector (has direction and sign) equal to dx/dt. Instantaneous speed is its magnitude, |dx/dt|, always positive or zero. They have the same numerical size at every instant.
How is instantaneous velocity related to the position-time graph?
It equals the slope of the tangent to the x-t graph at that instant. A steeper tangent means a larger speed; a tangent sloping downward means negative velocity.
Is dx/dt the same as ds/dt in these problems?
In one-dimensional NEET problems the position along the line is written as x or s, and v = dx/dt = ds/dt. Both symbols mean the position coordinate, so differentiate it to get velocity.
Why does NEET test instantaneous velocity so often?
Because it links calculus (differentiation) to motion and to graphs, and it hides the average-speed-vs-velocity trap. One clean step (v = dx/dt) unlocks many questions on same-velocity, reversal, and predicted-marks chapters.
