5. In this question i and j are perpendicular horizontal unit vectors.] A particle P is moving with constant acceleration . At 2pm, the velocity of p is (3i+5j)kmh^-1 and at 2.30 pm the velocity of P is (i+7j)kmh^-1 At time T hours after 2 pm, P is moving in the direction of the vector (-i+2j) (a) Find the value of T. Another particle,Q.has velocity V_(Q) km h^-1 at time t hours after 2 pm , where v_(Q)=(-4-2t)i+(mu +3t)j and u is a constant. Given that there is an instant when the velocity of P is equal to the velocity of 9, (b) find the value of mu (6)

To find the value of T, we need to use the concept of velocity and acceleration. Given that the particle P is moving with constant acceleration, we can find the acceleration vector by taking the difference between the final and initial velocities and dividing it by the time taken.The initial velocity at 2 pm is and the final velocity at 2.30 pm is . The time taken is 0.5 hours (30 minutes).So, the acceleration vector is: Simplifying this, we get: Now, we need to find the velocity of P at time T hours after 2 pm. We know that the velocity of P is in the direction of the vector . Let's represent the velocity of P as . Since the velocity is in the direction of , we can write: Simplifying this, we get: Now, we can use the equation of motion to find the velocity of P at time T: where is the initial velocity at 2 pm, which is .Substituting the values, we get: Simplifying this, we get: Now, we can solve for T by equating the coefficients of i and j on both sides of the equation: Using the relationship , we can substitute the values of x and y: Simplifying this, we get: Therefore, the value of T is hours.To find the value of , we need to equate the velocity of P with the velocity of Q.The velocity of P is given by: The velocity of Q is given by: Equating the velocities, we get: Simplifying this, we get: Using the relationship , we can substitute the values of x and y: Simplifying this, we get: Now, we need to find the value of t when the velocities are equal. We can do this by equating the magnitudes of the velocities: Simplifying this, we get: Solving this equation, we get: Substituting this value of T into the equation for , we get: Therefore, the value of is .