A projectile is accelerated from rest at point A along a circular track of radius 2 m, as shown in Fig. 5.1. Gravity acts in the z direction. The rate at which the speed of the projectile increases as a function of time, t, is at = 2t ms-2. What is the magnitude of the projectile’s total acceleration in the x-y plane when it is at point B?

To find the magnitude of the projectile's total acceleration in the x-y plane when it is at point B, we can break the acceleration into two components: radial acceleration and tangential acceleration.

1. Radial acceleration (ar):
Radial acceleration is the component of acceleration directed towards the center of the circular track. It is given by the formula ar = v² / r, where v is the velocity of the projectile and r is the radius of the circular track.

To find the velocity of the projectile at point B, we need to integrate the acceleration equation. Given that the initial velocity (v0) is 0, we can find the velocity equation by integrating the acceleration equation with respect to time (t):
v = ∫ (at) dt
= ∫ (2t) dt
= t² + C

Using the fact that the projectile is at point B, the time (t) it took to get from point A to B can be calculated. Let's denote it as tB. At tB, the position vector of the projectile is perpendicular to the tangent at point B.

The time it takes to reach point B can be determined by solving the equation r = v * tB, where r is the radius of the circular track and v is the velocity of the projectile at point B.

Plugging in the known values:
2m = (tB² + C) * tB

Since we know the initial position of the projectile (at point A), we can solve this equation for tB. Let's denote the solution as tB.

Now that we have the velocity at point B, we can calculate the radial acceleration:
ar = v² / rB
= (tB² + C)² / 2m

2. Tangential acceleration (at):
The tangential acceleration is the component of acceleration along the direction of motion. Given the acceleration as a function of time (at = 2t ms-2), we can calculate the tangential acceleration at point B by plugging in the corresponding value of tB:
atB = 2 * tB

3. Total acceleration (a):
The total acceleration is the vector sum of the radial and tangential accelerations. Since the radial acceleration is directed towards the center of the circular track and the tangential acceleration is directed along the direction of motion, we can find the magnitude of the total acceleration using the Pythagorean theorem:
aB = √(arB² + atB²)

Substituting the values found earlier, we can calculate the magnitude of the projectile's total acceleration at point B.