The velocity of a body A is given by v(t) = 2t^i + t
3^j. The body starts from rest at time t = 0 and
position s(0) = 2^i 10^j.
At what time, t, does the body cross the X axis?
What is the average velocity of the body for the time interval, t = 0 and the time that it crosses the
X axis?
What is the average acceleration of the body for the time interval, t = 0 and the time that it crosses
the X axis?
To find the time, t, at which the body crosses the X-axis, we know that the X-axis corresponds to the y-component of the velocity. So, we need to find the value of t for which the y-component of velocity, v(t), is equal to zero.
Given that v(t) = 2t^i + t3^j, we have v(t) = 0 at the time of crossing the X-axis. Solving for t, we equate the y-component of velocity to zero:
t3^j = 0
This implies that t = 0. Therefore, the body crosses the X-axis at t = 0.
To find the average velocity of the body over the time interval from t = 0 to the time it crosses the X-axis, we need to find the displacement of the body during that time.
The displacement, s, is obtained by integrating the velocity function with respect to time:
s(t) = ∫v(t) dt
For the given velocity function v(t) = 2t^i + t3^j, integrating with respect to time gives:
s(t) = t^2i + (1/2)t^2 3^j
Given that s(0) = 2^i 10^j, we can evaluate the displacement at t = 0:
s(0) = (0^2)i + (1/2)(0^2)3^j = 0^i + 0^j = 0
Therefore, the displacement at t = 0 is zero, and the average velocity over this time interval is also zero.
To find the average acceleration of the body over the same time interval, we need to find the change in velocity and divide it by the corresponding time interval.
The change in velocity, Δv, is given by the final velocity minus the initial velocity. In this case, the initial velocity is zero, as the body starts from rest.
Δv = v(t) - v(0) = 2t^i + t3^j - 0 = 2t^i + t3^j
The time interval, Δt, is the time it takes for the body to cross the X-axis, which we found earlier to be t = 0.
Δt = 0 - 0 = 0
Therefore, the average acceleration over this time interval is zero.