1. Time in the air:
When the ball is tossed upward, it reaches its maximum height and then falls back to the ground. The time it takes for the ball to reach its maximum height and fall back down is equal to the total time the ball is in the air.
To find the time in the air, we can use the equation for vertical motion:
h = u*t + (1/2)*g*t^2 ,
where:
h = height
u = initial velocity
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time
Since the ball starts from an initial height of 1.8 m and reaches the same height on its way back down, we can set the height (h) equal to 0:
0 = (21 m/s)*t + (1/2)*(9.8 m/s^2)*t^2
Rearranging the equation and solving for t using the quadratic formula, we get:
t = (-u ± sqrt(u^2 - 4*(1/2)*g*(0))) / (2*(1/2)*g)
= (-21 ± sqrt(21^2 - 4*(1/2)*9.8*(0))) / (2*(1/2)*9.8)
= (-21 ± sqrt(441)) / 9.8
Since we are only interested in the positive value of t, we can ignore the negative solution. Calculating t:
t = (-21 + sqrt(441)) / 9.8
= (-21 + 21) / 9.8
= 0 / 9.8
= 0
Therefore, the ball is in the air for 0 seconds.
2. Maximum height:
The maximum height reached by the ball can be found using the formula:
v^2 = u^2 + 2gh ,
where:
v = final velocity (0 m/s at maximum height)
u = initial velocity (21 m/s upward)
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height
Solving for h:
0^2 = (21 m/s)^2 + 2*(-9.8 m/s^2)*h
441 = 2*(-9.8)*h
h = 441 / (-19.6)
≈ -22.5 m
Since the height cannot be negative in this context, we discard the negative value. Therefore, the maximum height reached by the ball is approximately 22.5 meters above the ground.
3. Velocity when it hits the ground:
When the ball hits the ground, its velocity will be downward. The final velocity can be found using the equation:
v = u + gt ,
where:
v = final velocity
u = initial velocity (21 m/s upward)
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time in the air (0 seconds)
Plugging in the values:
v = 21 m/s + (9.8 m/s^2)*(0 s)
= 21 m/s
Therefore, the velocity that the ball hits the ground with is 21 m/s downward.