The position of an object is described as x = 4.0t and y= 1.5t2 - 2t + 1.0.
All factors are in SI-units.
a) What is the begining velocity of the object?
b) What is his acceleration?
c) What is the direction along the x-axe at the time t=1.0 s ?
Given answers are: 4.5 m/s in -27 degree; (0; 3.0 m/s2); +14 degree
dx/dt = 4
dy/dt = 3 t -2
so at t = 0
V = 4 i - 2 j
|V| = sqrt (16+9) = 5 m/s
theta = tan^-1 -2/4 = -26.6 degrees
b)
Ax = 0
Ay = 3 m/s^2
c) is a typo I think.
|V| = sqrt (16+4) = 4.47 m/s
theta = tan^-1 -2/4 = -26.6 degrees
Maybe in c you mean what is the velocity angle to the x direction at t = 1 ???
dx/dt = 4
dy/dt = 3-2 = 1
tan^-1 (1/4) = 14.03 degrees
Thank you very much!
You are welcome.
To find the answers to the given questions, we need to differentiate the equations of motion to find the velocities and accelerations:
a) Beginning velocity (v0) is the derivative of the position with respect to time (t):
v0 = dx/dt
Given: x = 4.0t
Differentiating x with respect to t:
v0 = d/dt(4.0t)
v0 = 4.0
Therefore, the beginning velocity of the object is 4.0 m/s.
b) Acceleration (a) is the derivative of velocity (v) with respect to time (t):
a = dv/dt
Given: y = 1.5t^2 - 2t + 1.0
Differentiating y with respect to t:
a = d/dt(1.5t^2 - 2t + 1.0)
a = 3.0t - 2.0
Therefore, the acceleration of the object is (3.0t - 2.0) m/s^2.
c) To find the direction along the x-axis at the time t = 1.0 s, we need to find the angle using trigonometry. The direction along the x-axis is given by the tangent of the angle.
Given: x = 4.0t
Substitute t = 1.0 s into the equation:
x = 4.0(1.0)
x = 4.0 m
Since the x-coordinate is positive (4.0 m), we need to find the angle with respect to the positive x-axis.
Using trigonometry, we can find the angle:
angle = arctan(y/x)
Substitute the values:
angle = arctan((1.5t^2 - 2t + 1.0)/4.0)
At t = 1.0 s:
angle = arctan((1.5(1.0)^2 - 2(1.0) + 1.0)/4.0)
angle = arctan(0.5/4.0)
Using a calculator, we find that arctan(0.5/4.0) is approximately 0.125.
Therefore, the direction along the x-axis at t = 1.0 s is approximately 0.125 radians, which is approximately 14 degrees.