Question

the particle is moving along x-axis in such a way so that its velocity versus time graph is
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Answer 1

A particle moves along x-axis in such a way that its x-coordinate varies with time according to the equation x = 4 - 2t - t². How will the speed of the particle vary with time?

Consider a particle moving on x axis. The distance travelled by the particle along the x axis is given by the function of time

x=4−2t−t2  

Rate of change of coordinate w.r.t time is dxdt=d(4−2t−t2)dt  

⇒dxdt=−2(1+t)  

Here, -ve sign indicates that the particle is moving towards the origin,  

When t=0 the particle is at, x=4−2×0−0=4⇒the particle is 4 units  

away from the origin  

At t=0 and at x=4 the speed of the particle w.r.t. coordinate of x axis is  

dxdt=−2(1+t)=−2  

Now, the acceleration of the particle can be found from the given function  

d2xdt2=d2(4−2t−t2)dt2=−2  

Here, -ve sign of the second derivative indicates the direction motion of the particle, which is towards the origin

When the x coordinate of the particle becomes 0? To know this roots of the function are found

⇒The roots of the function 0=4−2t−t2are,1.−3.2362.1.236  

Since time is not negative the value 1.236 is taken. Hence at t=1.236 units the particle reaches origin

For example when t=1 ; x=4−2t−t2=1units  

⇒ distance covered by the particle is 3 units  

The speed of the particle at 1 unit is −2(1+t)=−4 units per unit time  

When, t=3; x=11, and the total distance travelled fromt0tot3is 15 units  

This can be checked by using the equation s=ut+12(a)t2  

Data, u=−2,t=3,a—2  

|s|=|−2×3+12(−2)(3)2|  

s=|−6−9|=15units