You toss a tennis ball straight upward. At the moment it leaves your hand it is at a height of 1.3 m above the ground, and it is moving at a speed of 6.8 m/s.
(a) How much time does it take for the tennis ball to reach its maximum height?
s
(b) What is the maximum height above the ground that the tennis ball reaches?
m
(c) When the tennis ball is at a height of 2.2 m above the ground, what is its speed?
m/s
To answer these questions, we need to analyze the motion of the tennis ball.
(a) How much time does it take for the tennis ball to reach its maximum height?
To find the time it takes for the ball to reach its maximum height, we first need to determine its initial vertical velocity (v₀) when it leaves your hand. Since the ball is tossed straight upward, we know that the initial velocity is positive. In this case, v₀ = 6.8 m/s.
The ball will reach its maximum height when its vertical velocity becomes zero. At that point, it will start falling back down due to gravity. Using the kinematic equation for vertical motion:
vf = v₀ + at
where vf is the final velocity, v₀ is the initial velocity, a is the acceleration, and t is the time. In this case, the final velocity (vf) is 0 because the ball is momentarily at rest at its maximum height. The acceleration (a) is the acceleration due to gravity, which is approximately -9.8 m/s².
Setting vf = 0, we can solve for the time it takes for the ball to reach its maximum height:
0 = 6.8 m/s + (-9.8 m/s²)t
Rearranging the equation:
9.8 m/s²t = 6.8 m/s
t = 6.8 m/s / 9.8 m/s²
Calculating this, we get:
t = 0.694 seconds
Therefore, it takes approximately 0.694 seconds for the tennis ball to reach its maximum height.
(b) What is the maximum height above the ground that the tennis ball reaches?
To find the maximum height reached by the ball, we can use the kinematic equation for vertical motion:
s = v₀t + 1/2at²
where s is the displacement, v₀ is the initial velocity, t is the time, and a is the acceleration. Since the initial vertical velocity (v₀) is 6.8 m/s and the time (t) to reach the maximum height is 0.694 seconds, we can substitute these values into the equation:
s = 6.8 m/s * 0.694 seconds + 1/2 * (-9.8 m/s²) * (0.694 seconds)²
Calculating this, we get:
s ≈ 2.27 meters
Therefore, the maximum height above the ground that the tennis ball reaches is approximately 2.27 meters.
(c) When the tennis ball is at a height of 2.2 m above the ground, what is its speed?
To find the speed of the tennis ball when it is at a height of 2.2 meters above the ground, we can use the kinematic equation for vertical motion:
vf = v₀ + at
where vf is the final velocity, v₀ is the initial velocity, a is the acceleration, and t is the time. In this case, we need to find the final velocity when the ball is at a height of 2.2 meters, so vf is unknown. The initial velocity (v₀) is 6.8 m/s, and the acceleration (a) is -9.8 m/s².
Using the same equation as before and solving for vf:
vf = 6.8 m/s + (-9.8 m/s²)t
We need to find the time (t) it takes for the ball to reach a height of 2.2 meters. Using the equation we derived earlier:
2.2 m = 6.8 m/s * t + 1/2 * (-9.8 m/s²) * t²
We can rearrange and solve this quadratic equation to find the value of t. Using the quadratic formula or factoring, we get:
t = 1.58 seconds or t ≈ -0.3 seconds
Since time cannot be negative in this context, we discard the negative value. Therefore, it takes approximately 1.58 seconds for the ball to reach a height of 2.2 meters.
Now, substituting this value of t back into the initial equation:
vf = 6.8 m/s + (-9.8 m/s²) * 1.58 seconds
Calculating this, we get:
vf ≈ -9.24 m/s
Therefore, when the tennis ball is at a height of 2.2 meters above the ground, its speed is approximately 9.24 m/s downward (negative velocity indicates the downward direction).