A particle leaves its orgin with a velocity of 4 i m/s and a constant acceleration of (-2i + .5j) m/s^2. At the time the particle reaches its maximum X corrdinate a) what is its velocity? b) what is its position vector?
I'm not sure which equations i should use. I am confused by this problem
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To solve this problem, we can use the equations of motion for linear motion with constant acceleration:
1) v = u + at
2) s = ut + (1/2)at^2
3) v^2 = u^2 + 2as
Where:
u = initial velocity
v = final velocity
a = acceleration
s = displacement
t = time
In this problem, we are given the initial velocity (u = 4i m/s) and the acceleration (a = -2i + 0.5j m/s^2). The acceleration is given in vector form, where i and j are the unit vectors along the x and y axes respectively.
a) To find the final velocity (v) when the particle reaches its maximum x-coordinate, we need to determine the time at which this occurs. The maximum x-coordinate occurs when the particle comes to rest in the x-direction, i.e., when its x-component of velocity becomes zero.
Using equation 1), we have:
v_x = u_x + a_x t
0 = 4 - 2t
Solving for t, we get:
t = 2 seconds
Now, substituting the value of t into equation 1), we can find the final velocity (v):
v = u + at
v = 4i + (-2i + 0.5j) * 2
v = 0i + j
v = j m/s
Therefore, the velocity of the particle when it reaches its maximum x-coordinate is v = j m/s.
b) To find the position vector when the particle reaches its maximum x-coordinate, we need to find the displacement in the x-direction up to that point.
Using equation 2), we have:
s_x = u_x t + (1/2) a_x t^2
s_x = 4 * 2 + (1/2)(-2)(2)^2
s_x = 8 - 4
s_x = 4 meters
Since the particle starts from the origin, the x-coordinate component of the position vector is 4i meters.
The y-coordinate component of the position vector will be zero since the particle does not have any initial velocity or acceleration in the y-direction. Therefore, the y-coordinate component of the position vector is 0j meters.
Hence, the position vector when the particle reaches its maximum x-coordinate is s = 4i + 0j meters.