This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here's the solution to question 9: Given: Initial height, s_0 = 0.5 m Initial velocity, u = 16 m/s (upwards) Acceleration due to gravity, a = -10 m/s^2 (negative because it acts downwards, opposite to the initial upward velocity) a) an expression for its velocity after t seconds. Step 1: Integrate the acceleration to find the velocity expression. The acceleration is given by a = (dv)/(dt). v = a \, dt v = -10 \, dt v = -10t + C Step 2: Use the initial condition to find the constant of integration C. At t=0, the initial velocity is v = 16 m/s. 16 = -10(0) + C C = 16 So, the expression for velocity is: v = 16 - 10t \, m/s b) an expression for its height above the ground after t seconds. Step 1: Integrate the velocity expression to find the height (displacement) expression. The velocity is given by v = (ds)/(dt). s = v \, dt s = (16 - 10t) \, dt s = 16t - (10t^2)/(2) + D s = 16t - 5t^2 + D Step 2: Use the initial condition to find the constant of integration D. At t=0, the initial height is s = 0.5 m. 0.5 = 16(0) - 5(0)^2 + D D = 0.5 So, the expression for height above the ground is: s = 0.5 + 16t - 5t^2 \, m c) the maximum height reached by the ball. Step 1: Determine the time at which the maximum height is reached. Maximum height occurs when the velocity of the ball is momentarily zero (v=0). Set the velocity expression from part (a) to zero: 16 - 10t = 0 10t = 16 t = (16)/(10) = 1.6 \, s Step 2: Substitute this time into the height expression from part (b) to find the maximum height. s_max = 0.5 + 16(1.6) - 5(1.6)^2 s_max = 0.5 + 25.6 - 5(2.56) s_max = 0.5 + 25.6 - 12.8 s_max = 26.1 - 12.8 s_max = 13.3 \, m What's next?