A ball is thrown vertically upwards, from ground level, with an initial speed vo. Assume that air resistance is negligible so that the acceleration of the ball is due solely to gravity.
a) Derive an algebraic formula for the time t that it takes for the ball to reach its greatest height (as usual, use the symbol g for the magnitude of the acceleration of gravity).
b) Derive a formula for the maximum height h of the ball.
a) To derive the formula for the time it takes for the ball to reach its greatest height, we need to consider the motion of the ball under the influence of gravity.
At the highest point, the vertical velocity of the ball is zero because it momentarily stops before falling back down. Therefore, we can use the equation that relates the final velocity (vf), initial velocity (vo), acceleration (a), and time (t) as:
vf = vo + at
Since the ball is moving upwards and acceleration due to gravity is acting downwards, the gravitational acceleration (-g) should be used in this equation.
At the highest point, vf = 0, so we can rewrite the equation as:
0 = vo - gt
Rearranging the equation for t, we get:
gt = vo
t = vo / g
Therefore, the algebraic formula for the time it takes for the ball to reach its greatest height is:
t = vo / g
b) To derive the formula for the maximum height of the ball, we can use the kinematic equation that relates displacement (h), initial velocity (vo), time (t), and acceleration (a):
h = vo*t + (1/2)*a*t^2
Since we want to find the maximum height, we know that the final velocity (vf) at this point is zero. Substituting vf = 0 and using the equation for time from part a, we get:
h = vo * (vo/g) + (1/2)*(-g)*(vo/g)^2
Simplifying this equation, we have:
h = (vo^2)/(2g)
Therefore, the formula for the maximum height of the ball is:
h = (vo^2)/(2g)