Question 2:
If is the position of a particle in space at time .Find the angle between the velocity and acceleration vectors at time
To find the angle between the velocity vector and acceleration vector at a given time t, we need to determine the velocity and acceleration vectors at that time.
The velocity vector, V(t), represents the rate of change of position with respect to time. This can be calculated by taking the derivative of the position vector, R(t), with respect to time:
V(t) = dR(t)/dt
Similarly, the acceleration vector, A(t), represents the rate of change of velocity with respect to time. This can be calculated by taking the derivative of the velocity vector with respect to time:
A(t) = dV(t)/dt
Once we have the velocity and acceleration vectors at time t, we can find the angle between them by using the dot product. The dot product of two vectors A and B can be calculated as:
A · B = |A| |B| cosθ
Where |A| and |B| are the magnitudes of vectors A and B respectively, and θ is the angle between them.
In this case, we want to find the angle between V(t) and A(t) at time t. So, we calculate the dot product of V(t) and A(t), divide it by the product of their magnitudes, and take the inverse cos of the result:
θ = cos^(-1)((V(t) · A(t)) / (|V(t)| |A(t)|))
This will give us the angle between the velocity and acceleration vectors at the given time t.