Subjective Type

A projectile is launched with an initial speed of $$30 m/s$$ at an angle of $$60°$$ above the horizontal. What are the (a) magnitude and (b) angle of its velocity $$2.0 s$$ after launch, and (c) is the angle above or below the horizontal? What are the (d) magnitude and (e) angle of its velocity $$5.0 s$$ after launch, and (f) is the angle above or below the horizontal?

Solution

(a) We choose horizontal $$x$$ and vertical y axes such that both components of $$\vec { { v }_{ 0 } } $$ are positive.
Positive angles are counterclockwise from $$+x$$ and negative angles are clockwise from it.
In unit-vector notation, the velocity at each instant during the projectile motion is
$$\vec { v } ={ v }_{ 0 }{ cos\theta }_{ 0 }\hat { i } +({ v }_{ 0 }{ sin\theta }_{ 0 }-gt)\hat { j } .$$
putting values $${ v }_{ 0 }= 30 m/s $$ and $${ \theta }_{ 0 }= 60°$$,
$$\vec { v } =(15\hat { i } +6.4\hat { j } )m/s$$, for $$t = 2.0 s$$.
The magnitude of $$\vec { v } $$ is $$\left| \vec { v } \right| =\sqrt { { (15m/s) }^{ 2 }+{ (6.4m/s) }^{ 2 } } =16m/s.$$

(b) The direction of $$\vec { v } $$ is
$$\theta ={ tan }^{ -1 }[(6.4m/s)/(15m/s)]={ 23 }^{ o },$$
measured counterclockwise from $$+x$$.

(c) Since the angle is positive, it is above the horizontal.

(d) $$\vec { v } ={ v }_{ 0 }{ cos\theta }_{ 0 }\hat { i } +({ v }_{ 0 }{ sin\theta }_{ 0 }-gt)\hat { j } .$$
putting values $${ v }_{ 0 }= 30 m/s $$ and $${ \theta }_{ 0 }= 60°$$,
With $$ = 5.0 s,$$
$$\vec { v } =(15\hat { i } -23\hat { j } )m/s$$

(e) The direction of
$$\vec { v }$$ is $$\theta ={ tan }^{ -1 }[(-23m/s)/(15m/s)]={-57 }^{ o }$$,
i.e $$57°$$ measuredclockwise from $$+x$$.

(f) Since the angle is negative, it is below the horizontal.

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