a woman throws a ball vertically upwards so that 4seconds elapse before it strikes the ground. Determine the maximum height of the ball and the velocity with which the ball was thrown. Assume that the acceleration due to gravity is 10m/s squared and neglect the effect of air resistance
the height h(t) is
h(t) = vt-4.9t^2
h=0 when t=4, so
4v-4.9*16 = 0
v = 19.6
h(t) = 19.6t - 4.9t^2
max h at t=2 because of symmetry
h(2) = 19.6*2 - 4.9*4 = 19.6
To determine the maximum height and the velocity with which the ball was thrown, we can use the equations of motion. Here's how you can solve it step by step:
1. Determine the time it takes for the ball to reach its maximum height:
Since the ball takes 4 seconds to strike the ground, half of this time (2 seconds) is the time it takes to reach the maximum height. Note that the time measured from the beginning (when the ball was thrown) is different from the time measured from the peak height.
2. Calculate the velocity at the maximum height using the equation of motion:
The equation to find the velocity at any given time is: v_f = v_i + a*t, where v_f is the final velocity, v_i is the initial velocity, a is the acceleration, and t is the time interval.
At the maximum height, the final velocity is zero, and the acceleration due to gravity is acting in the opposite direction of the initial velocity. Therefore, we have: 0 = v_i - 10*2. Solve this equation to find the initial velocity.
3. Calculate the maximum height using the equation for displacement:
The equation to find displacement (height) is: s = v_i*t + (1/2)*a*t^2, where s is the displacement, v_i is the initial velocity, a is the acceleration, and t is the time.
Since the ball reaches its maximum height after 2 seconds, we can use this equation with the initial velocity found in step 2 and the upward acceleration due to gravity. Set the equation equal to zero: 0 = v_i*(2) + (1/2)*(-10)*(2)^2 and solve for the maximum height.
By following these steps, you can determine the maximum height of the ball and the velocity with which it was thrown.