a tennis ball was thrown vertically upwards with an initial velocity of 20m/s calculate its velocity after 4seconds, it maximum height attained and time taken to reach it
h(t)=vi*t-4.8t^2
v(t)=vi + g t
at max height, v(t)=0, solve for t when that happens.
To calculate the tennis ball's velocity after 4 seconds, its maximum height attained, and the time it takes to reach the maximum height, we can use the equations of motion for an object thrown vertically upwards.
1. Velocity after time t:
The velocity of the ball after a certain time "t" can be calculated using the equation:
v = u + at
where:
v = final velocity
u = initial velocity
a = acceleration
t = time
Since the tennis ball is being thrown vertically upwards, the acceleration is equal to the acceleration due to gravity, which is approximately -9.8 m/s² (negative because it acts in the opposite direction of the initial velocity).
Given:
u = 20 m/s (initial velocity)
t = 4 s (time)
Using the equation, we can calculate the velocity after 4 seconds:
v = 20 + (-9.8) * 4
v = 20 - 39.2
v ≈ -19.2 m/s
Therefore, the velocity of the tennis ball after 4 seconds is approximately -19.2 m/s. The negative sign indicates that the ball is moving downwards.
2. Maximum height attained:
To calculate the maximum height attained by the tennis ball, we need to know the time at which it reaches its highest point. The time taken to reach the maximum height is the same as the time taken to fall back down to the point of projection (considering air resistance to be negligible).
Since the ball is thrown vertically upwards, its total time of flight can be calculated using the equation:
t_total = 2 * t (time of ascent is equal to time of descent)
Given:
t = 4 s (time)
Calculating the total time of flight:
t_total = 2 * 4
t_total = 8 s
Now, we can calculate the maximum height attained using the equation for displacement:
s = ut + (1/2) * a * t²
where:
s = displacement
u = initial velocity
a = acceleration
t = time
Since the ball reaches its highest point when its velocity becomes 0, we can use this equation to find the displacement at that time:
0 = 20 + (-9.8) * t_max
t_max = 20 / 9.8
t_max ≈ 2.04 s
Now, calculating the maximum height:
s_max = u * t_max + (1/2) * a * t_max²
s_max = 20 * 2.04 + (0.5) * (-9.8) * (2.04)²
s_max ≈ 20.4 m
Therefore, the maximum height attained by the tennis ball is approximately 20.4 meters.
3. Time taken to reach the maximum height:
We have already calculated the time taken to reach the maximum height as approximately 2.04 seconds.
To summarize:
- The velocity of the tennis ball after 4 seconds is approximately -19.2 m/s.
- The maximum height attained by the tennis ball is approximately 20.4 meters.
- The time taken to reach the maximum height is approximately 2.04 seconds.