You toss a tennis ball straight upward. At the moment it leaves your hand it is at a height of 1.5 m above the ground, and it is moving at a speed of 8.0 m/s.

(a) How much time does it take for the tennis ball to reach its maximum height?

(b) What is the maximum height above the ground that the tennis ball reaches?

(c) When the tennis ball is at a height of 2.4 m above the ground, what is its speed?

Am I supposed to use the free fall formula?

Oh, tennis balls and gravity, what a smashing combination! Yes, you can use the good ol' free fall formula to solve these questions. Let's get juggling!

(a) To find the time it takes for the tennis ball to reach its maximum height, we can use the formula 𝑣 = 𝑒 + π‘Žπ‘‘, where 𝑣 is the final velocity, 𝑒 is the initial velocity, π‘Ž is the acceleration, and 𝑑 is the time. Since the ball is going upwards, its acceleration is a constant -9.8 m/sΒ² (we'll call it 𝑔 for giggles), and the final velocity (𝑣) is 0 m/s when it reaches its maximum height. So, we have 0 = 8 - 9.8𝑑. Solve it for 𝑑, and you'll find the flight time.

(b) The maximum height reached is a classic! You can use the equation 𝑠 = 𝑒𝑑 + 0.5π‘Žπ‘‘Β², where 𝑠 is the displacement, 𝑒 is the initial velocity, 𝑑 is the time, and π‘Ž is the acceleration. Plug in the values and solve for 𝑠 to find the highest point the tennis ball reaches.

(c) Ah, the speed at a specific height! We can use the equation 𝑣² = 𝑒² + 2π‘Žπ‘ . Now, 𝑣 is the unknown we're looking for, 𝑒 is the initial velocity, π‘Ž is the acceleration, and 𝑠 is the displacement (height in this case). Substitute the given values and solve for 𝑣 to discover the speed at that particular height.

It may seem daunting, but fear not! Grab your juggling balls, put on your clown makeup, and solve those equations with a smile!

Yes, you can use the free fall formula to solve these problems since the ball is only influenced by gravity while it's in the air. The free fall formula for the vertical motion of an object is:

v = u + gt

Where:
v = final velocity
u = initial velocity
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time taken

Let's solve the problem step by step:

(a) To find the time it takes for the ball to reach its maximum height, we need to find the initial velocity (u). The given initial velocity is 8.0 m/s. Since the ball is moving vertically upward, the final velocity when it reaches its maximum height will be zero (v = 0 m/s). We can plug these values into the formula:

0 = 8.0 m/s + (-9.8 m/s^2) * t

Simplifying the equation, we have:

-8.0 m/s = -9.8 m/s^2 * t

To isolate t, divide both sides of the equation by -9.8 m/s^2:

t = (-8.0 m/s) / (-9.8 m/s^2)

t β‰ˆ 0.82 seconds

So, it takes approximately 0.82 seconds for the tennis ball to reach its maximum height.

(b) The maximum height can be determined using the formula for displacement:

s = ut + 0.5gt^2

Since we know the initial velocity (u), time (t), and acceleration due to gravity (g), we can plug the values into the formula:

s = (8.0 m/s) * 0.82 s + 0.5 * (-9.8 m/s^2) * (0.82 s)^2

Simplifying the equation, we have:

s β‰ˆ 3.3 meters

Therefore, the maximum height reached by the tennis ball is approximately 3.3 meters above the ground.

(c) To find the speed of the tennis ball when it is at a height of 2.4 meters, we can use the same formula as in part (b), but solve for the final velocity (v) instead:

s = ut + 0.5gt^2

Rearranging the formula:

v^2 = u^2 + 2gs

Plugging in the values:

v^2 = (8.0 m/s)^2 + 2 * (-9.8 m/s^2) * 2.4 m

Simplifying the equation, we have:

v β‰ˆ 5.8 m/s

Therefore, when the tennis ball is at a height of 2.4 meters, its speed is approximately 5.8 m/s.

No, you should not use the free fall formula for this problem because the ball is not in free fall. Instead, you should use the equations of motion for an object undergoing uniform acceleration, specifically the equations for motion under constant acceleration in the vertical direction.

To answer the first question:

(a) How much time does it take for the tennis ball to reach its maximum height?
We can use the equation for displacement:

Ξ”y = v0t + (1/2)atΒ²

where Ξ”y is the change in height, v0 is the initial velocity, t is time, and a the acceleration due to gravity (-9.8 m/sΒ²).

Since the ball is tossed straight upward, we can take the initial velocity (v0) as positive (+8.0 m/s).

The desired displacement is Ξ”y = 0 m (because the ball reaches its maximum height, it stops moving upwards). So we have:

0 = (8.0 m/s) t + (1/2)(-9.8 m/sΒ²)tΒ²

Solving this quadratic equation for t will give you the time it takes for the ball to reach its maximum height.

To answer the second question:

(b) What is the maximum height above the ground that the tennis ball reaches?
Once you find the time (t) from the first question, you can substitute it into the equation for displacement to find the maximum height. The equation is:

Ξ”y = v0t + (1/2)atΒ²

where v0 is still +8.0 m/s and a is -9.8 m/sΒ². Plug in the known values and solve for Ξ”y.

To answer the third question:

(c) When the tennis ball is at a height of 2.4 m above the ground, what is its speed?
To find the speed, we need to use the equation for velocity:

v = v0 + at

where v0 is the initial velocity (+8.0 m/s), a is the acceleration due to gravity (-9.8 m/sΒ²), and v is the velocity (which we want to find).

Since the ball is moving upward when it reaches a height of 2.4 m, the final velocity (v) would be negative (-). So substitute the known values into the equation and solve for v.

v = Vi - 9.81 t

at top, v = 0
t = 8 / 9.81

(b) h = 1.5 + 8 t -(1/2)(9.81) t^2

(c) at 2.4 m, new t

2.4 - 1.5 = 8 t - 4.9 t^2
then v = 8 - 9.81 t