Velocity Time Graphs

Description:

The Concept

Velocity – Time graphs display the motion of a particle by showing the changes of velocity with respect to time.

For Example,

Above velocity – time graph represents an object moving with uniform velocity, v₁, in positive direction.

Above velocity – time graph represents particle going at uniform velocity in positive direction until, t = t₁, when it changes direction and starts moving with uniform velocity, v₁, in opposite direction.

Sailent Points

Velocity – Time graphs are also called v − t graphs.

Slope of Velocity – Time graph gives instantaneous acceleration.

• Slope of any curve = dy/dx (Revisit Differential Calculus for more details)

• Slope of Velocity - time graph = dx/dt = inst.acceleration

Area under the Velocity – Time graph gives Displacement (or Change in Position).

• Area under any curve = x₂∫x₁ f(x) . dx

• Area under Velocity - time graph = t₂∫t₁ v . dt = displacement

Example Problem 1

Calculate the acceleration of the object in each interval of time.

Solution

Velocity in the first 5s, a = dv/dt = 20 - 0/5 - 0 m/s² = 4 m/s²

Velocity in between 5s and 10s, v = dx/dt = 20 - 20/10 - 5 m/s² = 0 m/s²

Velocity in between 10s and 15s, v = dx/dt = 0 - 20/15 - 10 m/s² = -4 m/s²

Example Problem 2

Find the final position of the object if it started at x = 5m.

We know area under the curve of v − t graph gives displacement.

Area = 1/2 × 20 × 5 + 20 × 5 + 1/2 × 20 × 5 = 200 m

Therefore,

Final Position = Initial Position + Displacement

Final Position = 5 + 200 = 250 m

Note −

• A line parallel to time axis indicates constant velocity motion.

• Line parallel to velocity axis is not possible. It would indicate particle changing its velocity in no time.