Question

A particle moves along x-axis with an initial speed v₀ = 5 ms^-1. If its acceleration varies with time as shown in a-t graph in figure,
(a) Find the velocity of the particle at t=4s.
(b) Find the time when the particle starts moving along -x direction.

Answer 1

The velocity of the particle at t = 4s can be given as

\(\vec{v}_4=\vec{v}_0+△\vec{v}\,\,..(i)\)

where \(△\vec{v}=\) A

(= area under a-t graph during first four seconds)

Referring to a-t graph, we have    ...(ii)

where A₁​ = 5 x 1 = 5, A₂​ = (1/2) x x x 5,

A₃​ = (1/2) x (1 - x) x 10, and A₄​ = (1/2) x 2 x 10 = 10

We can find x as following:

Using properties of similar triangles, we have \(\frac{x}{5}=\frac{1-x}{10}\)

This yields x = 1/3.

Substituting x = (1/3) in A₂​ and A_3​

we have A₂​ = 5/6 and A₃​ = 10/3.

Then substituting A₁​, A₂​, A₃​ and A₄​ in (ii), we have A = -7.5.

Negative area tells us that change in velocity is along -x direction

\(△\vec{v}=-7.5\,m/s\)

Hence substituting in (1), \(△\vec{v}_0=5\,m/s\) and \(△\vec{v}= -7.5\,m/s\)

we have \(△\vec{v}_4=\vec{v}_0+△\vec{v}=5-7.5=-2.5\,m/s\)