Question

Entering his dorm room, a student tosses his book bag to the right and upward at an angle of$\mathbf{45}\mathbf{°}$ with the horizontal (Fig. OQ4.2). Air resistance does not affect the bag. The bag moves through point (A) immediately after it leaves the student's hand, through point (B) at the top of its flight, and through point (C) immediately before it lands on the top bunk bed.

(i) Rank the following horizontal and vertical velocity components from the largest to the smallest. (a)${\mathbf{v}}_{\mathbf{a}\mathbf{x}}$ (b)${\mathbf{v}}_{\mathbf{a}\mathbf{y}}$ (c) ${\mathbf{v}}_{bx}$(d)${\mathbf{v}}_{\mathbf{a}y}$ (e)${\mathbf{v}}_{\mathbf{c}\mathbf{y}}$ Note that zero is larger than a negative number. If two quantities are equal, show them as equal in your list. If any quantity is equal to zero, show that fact in your list.

(ii) Similarly, rank the following acceleration components.

(a) ${\mathit{a}}_{\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{x}}$

(b) ${\mathit{a}}_{\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{y}}$

(c) ${\mathit{a}}_{\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{x}}$

(d)${\mathit{a}}_{\mathbf{\text{(B)}}\mathbf{y}}$

(e)${\mathit{a}}_{\mathbf{\left(}\mathbf{C}\mathbf{\right)}{\mathbf{y}}^{\mathbf{*}}}$

Short answer

(i) The velocity components are ranked as

$v_{{\left(A\right)}_x}=v_{{\left(B\right)}_x}=v_{{\left(C\right)}_x}=v_{{\left(A\right)}_y}>v_{{\left(B\right)}_y}>v_{{\left(C\right)}_y}$

(ii)The rank of acceleration components

$a_{{\left(A\right)}_x}=a_{{\left(B\right)}_x}=a_{{\left(C\right)}_x}>a_{{\left(A\right)}_y}=a_{{\left(B\right)}_y}=a_{{\left(C\right)}_y}$

Step-by-step solution

Step 1: Define magnitude and direction of the vector

The magnitude of vector is represented by its length. The direction of a vector is represented by its unit vector. Unit vectors are vectors having unit magnitude.

Now calculate x and y component

The horizontal component of velocity remains constant during the motion of the bag projectile.

Then$v_{{\left(A\right)}_x}=v_{\left(B\right)x}=v_{\left(C\right)x}$

At point (A)$v_{{\left(A\right)}_x}=v_{{\left(A\right)}_y}$

At point (B)(maximum height) $v_{{\left(B\right)}_y}=0$

At point $\left(C\right),v_{{\left(C\right)}_y}$ has some negative values

Then$v_{{\left(A\right)}_x}=v_{{\left(B\right)}_x}=v_{{\left(C\right)}_x}=v_{{\left(A\right)}_y}>v_{{\left(B\right)}_y}>v_{{\left(C\right)}_y}$

Now find acceleration in projectile

ii) There is acceleration in projectile motion, which is due to gravity, but there will be no acceleration in the x direction.

Then

$a_{{\left(A\right)}_x}=a_{{\left(B\right)}_x}=a_{{\left(C\right)}_x}=0$

$a_{{\left(A\right)}_y}=a_{{\left(B\right)}_y}=a_{{\left(C\right)}_y}=-g$

The acceleration can be ranked as-

$a_{{\left(A\right)}_x}=a_{{\left(B\right)}_x}=a_{{\left(C\right)}_x}>a_{{\left(A\right)}_y}=a_{{\left(B\right)}_y}=a_{{\left(C\right)}_y}$