A car starts from rest and accelerates at \( 2 \, \text{ms}^{-2} \) for 5 seconds. Find the final velocity of the car and the distance traveled.

Known: \( a = 2\,\text{ms}^{-2}, t = 5\,s, u = 0\,\text{ms}^{-1} \)

\( v = u + at = 0 + 2 \times 5 = 10\,\text{ms}^{-1} \)

\( s = ut + \frac{1}{2}at^2 = 0 + \frac{1}{2} \times 2 \times 25 = 25\,\text{m} \)

A ball is thrown vertically upward with an initial velocity of \( 30\,\text{ms}^{-1} \).

\( a = -10\,\text{ms}^{-2}, u = 30\,\text{ms}^{-1}, v = 0 \)

\( s = \frac{v^2 - u^2}{2a} = \frac{0 - 900}{-20} = 45\,\text{m} \)

\( t = \frac{v - u}{a} = \frac{-30}{-10} = 3\,\text{s} \)

Total time of flight = \( 2 \times 3 = 6\,s \)

A cyclist moving at 5 m/s starts accelerating at 1 m/s² for 10 seconds, then decelerates at 0.5 m/s² until coming to rest. Find the total distance and total time.

\( s_1 = ut + \frac{1}{2}at^2 = 5 \times 10 + \frac{1}{2} \times 1 \times 100 = 100\,\text{m} \)

\( v = u + at = 5 + 10 = 15\,\text{ms}^{-1} \)

\( s_2 = \frac{-v^2}{2a} = \frac{-225}{-1} = 225\,\text{m} \)

\( t = \frac{-v}{a} = \frac{-15}{-0.5} = 30\,s \)

Total distance = \( 100 + 225 = 325\,\text{m} \), total time = \( 10 + 30 = 40\,s \)

A ball is launched with \( 20\,\text{ms}^{-1} \) at \( 45^\circ \). Find the range, max height, and total flight time.

\( v_x = 20 \cos(45) = 14.14\,\text{ms}^{-1} \), \( v_y = 20 \sin(45) = 14.14\,\text{ms}^{-1} \)

\( s = \frac{-v_y^2}{2a} = \frac{-200}{-19.6} = 10.2\,\text{m} \)

\( t = \frac{-v_y}{a} = \frac{-14.14}{-9.8} = 1.44\,s \), total = \( 2.88\,s \)

\( x = v_x \cdot t = 14.14 \times 2.88 = 40.72\,\text{m} \)