(III) A particle revolves in a horizontal circle of radius 1.95 m. At a particular instant, its acceleration is 1.05m/s^2 in a direction that makes an angle of 25.0^circ to its direction of motion . Determine its speed (a) at this moment, and (b) 2.00 s later ,assumin g constant tangential acceleration.

To solve this problem, we need to use the relationship between centripetal acceleration, tangential acceleration, and the velocity of the particle.<br /><br />Given:<br />- Radius of the circle, $r = 1.95 \, \text{m}$<br />- Centripetal acceleration, $a_c = \frac{v^2}{r}$<br />- Tangential acceleration, $a_t = a \sin(\theta)$<br />- Angle, $\theta = 25.0^\circ$<br /><br />(a) To find the speed of the particle at the given instant, we can use the formula for centripetal acceleration:<br /><br />$a_c = \frac{v^2}{r}$<br /><br />We also know that the total acceleration is given by:<br /><br />$a = \sqrt{a_c^2 + a_t^2}$<br /><br />Since $a_c = \frac{v^2}{r}$ and $a_t = a \sin(\theta)$, we can solve for $v$:<br /><br />$v = \sqrt{a_c \cdot r}$<br /><br />Substituting the given values:<br /><br />$a_c = 1.05 \, \text{m/s}^2$<br /><br />$r = 1.95 \, \text{m}$<br /><br />$v = \sqrt{1.05 \cdot 1.95}$<br /><br />$v \approx 1.10 \, \text{m/s}$<br /><br />(b) To find the speed of the particle 2.00 s later, we need to consider the tangential acceleration. The tangential acceleration is given by:<br /><br />$a_t = a \sin(\theta)$<br /><br />Substituting the given values:<br /><br />$a = 1.05 \, \text{m/s}^2$<br /><br />$\theta = 25.0^\circ$<br /><br />$a_t = 1.05 \cdot \sin(25.0^\circ)$<br /><br />$a_t \approx 0.45 \, \text{m/s}^2$<br /><br />The tangential velocity after 2.00 s is given by:<br /><br />$v_t = v + a_t \cdot t$<br /><br />Substituting the given values:<br /><br />$v = 1.10 \, \text{m/s}$<br /><br />$a_t = 0.45 \, \text{m/s}^2$<br /><br />$t = 2.00 \, \text{s}$<br /><br />$v_t = 1.10 + 0.45 \cdot 2.00$<br /><br />$v_t \approx 2.00 \, \text{m/s}$<br /><br />Therefore, the speed of the particle at the given instant is approximately $1.10 \, \text{m/s}$, and the speed of the particle 2.00 s later is approximately $2.00 \, \text{m/s}$.