Question

2. The velocity vector of a particle is v(t) (20+4.0t) i- 15jm/s What is the vector in component form and its speed at t-5s? 3. A golf ball is hit off a tee at the edge of a cliff has position vector r(t) = 18.0t i + (4.00t-4.90t*) j m What is the speed of the ball at t 3.00 s? 4. Extra Credit: Find the velocity vector and speed at t 5 of a particle whose acceleration components are as follows 1 234 5 1 2 3 4 5

plz write out solution I have a hard time understanding typed out solutions thanks

Answer 1

NOTE :  

(1) A Vector can be represented in three forms :

(i) Graphical form

Example : -

(ii) Component form :

Example:

(iii) Polar form

In this form, we write the vector by writing its magnitude and the angle made by the vector with positive x - axis in anticlockwise direction .

The vector r can be written as ;

Where, $r=$ is magnitude of the vector r and

is the angle made by the vector with positive x -axis in anticlockwise direction

The x - component will be  

And the y - component will be

For the example considered above, and

So,

(B) If the two points on a straight line is known then the equation of the straight line can be written as

Where, (x₁ , y₁) and (x₂ , y₂ ) are the two points on the straight line.  

QUESTIONS :

(2) Given,

Now, at t = 5 s :

or,   

This is the component form of the given vector at t= 5 seconds. In which the x - component of the vector is 40 m/s (i.e. velocity/speed of the particle in x direction ) and the y - component is -15 m/s (the negative sign is indicating that the velocity is in negative y direction). This vector can also be written as

We know that, the magnitude of the velocity vector gives the speed of the particle.  

This is the speed of the particle at t = 5 seconds.

(3) Given, the position vector of the golf ball is

We know that the velocity is given by

  

So,

V(t)(18.0(4.00t 4.902)) 18.0(4.00-9.80t)j m/s dt

At t = 3 seconds

  

So,  

  

This is the speed of the golf ball at t = 3 seconds.

(4)  According to the given plots,

In a_x Vs t plot (i.e. the plot of x component of acceleration), the equation of the line can be written as

  

or,

or,   

And, we know that

or,  

or,

Similarly, from the plot of a_y Vs t (i.e. the plot of y component of acceleration) , we can write the equation of line as

(as the line is parallel to the t - axis)

So,

So, we can write velocity vector as

Now, at t = 5 seconds

   in m/s

So,  

SpeedVt)- V152 + 152-15V2 m/s

Questions in the second page :

(4) Vectors in the polar form :

(a) Given vector is

For this vector,  

and   

So,   

(b) Given vector is

For this vector,

and

So,   

(5V2, 45%

For any doubt please comment and please give an up vote. Thank you.