Class 11 Physics · Chapter 2 NotesMotion in a Straight Line

To describe motion we first fix a reference point called the origin and choose a positive direction along the line of motion. The position of an object is its coordinate with respect to this origin; positions to the right are taken as positive and to the left as negative. The path length is the total distance actually covered, while displacement is the change in position, written as Δx = x₂ – x₁. Displacement has both magnitude and direction, so it can be positive, negative or zero, whereas path length is always positive. For motion along a straight line without any reversal, the magnitude of displacement equals the path length; if the object turns back, the path length becomes greater.

• Displacement Δx = x₂ – x₁ (a vector quantity in one dimension, with sign)
• Path length = total distance travelled (a scalar, always positive)
• Magnitude of displacement ≤ path length

Average velocity tells how fast position changes over a time interval. It is the displacement divided by the time interval, so it carries the sign of the displacement. Average speed is the total path length divided by the time interval and is always positive. Because path length is at least as large as the magnitude of displacement, average speed is greater than or equal to the magnitude of average velocity over any finite interval. The two become equal when the object moves in one direction without reversing. Both are measured in metres per second (m s⁻¹) in SI units.

• Average velocity = Δx / Δt
• Average speed = total path length / Δt
• Average speed ≥ magnitude of average velocity

Average velocity over a long interval hides how fast the object moved at each instant. To capture this, we shrink the time interval Δt to be infinitesimally small. The limiting value of Δx/Δt as Δt tends to zero is the instantaneous velocity, written v = dx/dt. On a position-time graph, this equals the slope of the tangent drawn at that instant. Instantaneous speed is simply the magnitude of instantaneous velocity, so a velocity of +24 m s⁻¹ and one of –24 m s⁻¹ both correspond to a speed of 24 m s⁻¹. For uniform motion, velocity is the same at every instant and equals the average velocity.

• v = dx/dt (limit of Δx/Δt as Δt → 0)
• Slope of the tangent to the x-t graph gives instantaneous velocity
• Instantaneous speed = magnitude of instantaneous velocity

Velocity usually changes during motion, and the rate of change of velocity with time is called acceleration. Average acceleration over an interval is the change in velocity divided by the time interval, a = Δv/Δt. Instantaneous acceleration is the limit of this ratio as Δt tends to zero, a = dv/dt, and equals the slope of the tangent to the velocity-time graph. Acceleration may arise from a change in speed, a change in direction, or both. Its SI unit is m s⁻². A useful result is that the area under the velocity-time curve between two instants equals the displacement during that interval.

• Average acceleration a = Δv/Δt
• Instantaneous acceleration a = dv/dt
• Area under the v-t curve = displacement

When acceleration is constant, the average acceleration equals the constant value a, and simple relations connect the five quantities: displacement x, time t, initial velocity v₀, final velocity v and acceleration a. Starting from v = v₀ + at and using the area under the v-t graph, we obtain x = v₀t + ½at² and v² = v₀² + 2ax. These are the kinematic equations of rectilinear motion for constant acceleration. If the object starts at x₀ instead of the origin, x is replaced by (x – x₀). All quantities are algebraic, so signs must be substituted carefully.

• v = v₀ + at
• x = v₀t + ½at²
• v² = v₀² + 2ax
• If position at t = 0 is x₀, replace x by (x – x₀)

An object released near the Earth's surface, with air resistance neglected, is in free fall. It moves with a constant downward acceleration g, about 9.8 m s⁻². Taking upward as positive, a = –g, and the kinematic equations become v = –gt, y = –½gt² and v² = –2gy for an object released from rest at y = 0. Galileo showed that the distances covered in successive equal time intervals by a body falling from rest are in the ratio 1 : 3 : 5 : 7 : 9 : 11 …, known as the law of odd numbers. Free fall is therefore a standard example of uniformly accelerated motion.

• g ≈ 9.8 m s⁻², directed downward
• For free fall from restv = –gt, y = –½gt², v² = –2gy (upward positive)
• Distances in successive equal intervals1 : 3 : 5 : 7 …

Motion is always described with respect to a chosen frame of reference, so the velocity of an object depends on the observer. If two objects A and B move along the same straight line with velocities v_A and v_B, the velocity of A relative to B is v_AB = v_A – v_B, and the velocity of B relative to A is v_BA = v_B – v_A. When both move in the same direction, the relative velocity is the difference of speeds; when they move in opposite directions, it is the sum. This idea explains everyday situations such as two trains passing each other or a boat moving in a river.

• v_AB = v_A – v_B
• v_BA = –v_AB
• Same directionsubtract speeds; opposite directions: add speeds