a girl throws a ball vertically upward with a speed of 20 ft/sec from the roof of a building 60 ft high.(a)how long will it take the ball to reach the ground? (b)what will its speed be when it strikes the ground?
a. V = Vo + g*Tr = 0, Tr = -Vo/g = -20/-32 = 0.625 s. = Rise time.
Tf = Tr = 0.625 s. = Fall time from max ht. to top of bldg.
h = Vo*T + 0.5g*T^2. h = 60 Ft., Vo = 20 Ft/s, g = 32 Ft/s^2, T = ?.
Tr+Tf+T = Time to reach gnd.
b. V^2 = Vo^2 + 2g*h. Vo = 20 Ft/s, g = 32 Ft/s^2, h = 60 Ft., V = ?.
To find the time it takes for the ball to reach the ground and its speed when it strikes the ground, we can use the laws of motion and kinematic equations.
(a) To find how long it takes for the ball to reach the ground, we can use the equation:
h = u*t - (1/2) * g * t^2
Where:
h = height or displacement (in this case, it is the distance the ball falls, which is 60 ft)
u = initial velocity (20 ft/sec)
g = acceleration due to gravity (32.2 ft/sec^2, since we are dealing with feet)
Plugging in the values, we get:
60 = 20*t - (1/2) * 32.2 * t^2
Simplifying the equation, we have:
0 = 16.1t^2 - 20t + 60
Now we can solve this quadratic equation for t by factoring, completing the square, or using the quadratic formula. In this case, using the quadratic formula seems most convenient.
The quadratic formula is:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation 0 = 16.1t^2 - 20t + 60, a = 16.1, b = -20, and c = 60.
Using the quadratic formula, we can substitute these values to calculate the two possible roots for t. However, we know that the negative root will not be meaningful in this context since we are looking for the time it takes for the ball to reach the ground (a positive quantity). Therefore, we only need to consider the positive root.
Once you calculate the positive root for t, you will have the time it takes for the ball to reach the ground.
(b) To find the speed of the ball when it strikes the ground, we can use the kinematic equation:
v = u - g * t
Where:
v = final velocity (which is the speed when the ball strikes the ground)
u = initial velocity (20 ft/sec)
g = acceleration due to gravity (32.2 ft/sec^2)
t = time calculated in part (a)
Substituting the values, we get:
v = 20 - 32.2 * t
Evaluating the value of t from part (a) and substituting it into the equation, you will get the speed of the ball when it strikes the ground.