When a 58 gram tennis ball is served, it accelerates from rest to a speed of 45 m/s. The impact with the racket gives the ball a constant acceleration over a distance of 44 cm. What is the magnitude of the net force acting on the ball?
Well, if you want to know the magnitude of the net force acting on the ball, you've come to the right party! Let's calculate it together, shall we?
First, we need to find the time it takes for the ball to reach a speed of 45 m/s. We can use the formula v = u + at, where v is the final velocity (45 m/s), u is the initial velocity (0 m/s), and a is the acceleration. Rearranging the equation, we have t = (v - u) / a.
Plugging in the values, we get t = (45 m/s - 0 m/s) / a. But, wait! We haven't calculated the acceleration yet! Silly me! To find the acceleration, we can use the equation a = (v^2 - u^2) / (2s), where s is the distance (44 cm) and u is the initial velocity.
Converting the distance to meters, we have s = 44 cm = 0.44 m. Now we can calculate the acceleration. Plugging in the values, we get a = (45 m/s)^2 - (0 m/s)^2 / (2 * 0.44 m).
Doing the math, we have a = 45^2 m^2/s^2 / 0.88 m = 2025 m^2/s^2 / 0.88 m = 2306.82 m/s^2.
Now that we know the acceleration, we can calculate the time it takes for the ball to reach 45 m/s. Plugging the acceleration into our previous equation, we have t = (45 m/s - 0 m/s) / 2306.82 m/s^2.
Doing the fancy math, we get t = 0.0195 s (approximately).
Now, let's find the net force! We can use the equation F = ma, where m is the mass of the ball (58 g = 0.058 kg) and a is the acceleration.
Substituting our values, we have F = 0.058 kg * 2306.82 m/s^2.
Calculating the force, we get F = 133.64 N (approximately).
So, the magnitude of the net force acting on the ball is approximately 133.64 Newtons. It's quite a smashing force, I must say!
To find the magnitude of the net force acting on the ball, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration.
First, let's convert the given values into the correct units. The mass of the ball is 58 grams, which is equal to 0.058 kilograms. The distance over which the ball experiences constant acceleration is 44 cm, which is equal to 0.44 meters.
Next, we need to find the acceleration of the ball. We can use the equation of motion that relates acceleration, initial velocity, final velocity, and displacement:
v^2 = u^2 + 2as
In this case, the initial velocity (u) is 0 because the ball starts from rest, the final velocity (v) is 45 m/s, and the displacement (s) is 0.44 meters. Plugging these values into the equation, we have:
45^2 = 0^2 + 2a(0.44)
2025 = 0.88a
Solving for acceleration (a), we get:
a = 2025 / 0.88
a ≈ 2306.82 m/s^2
Now that we know the mass of the ball (m = 0.058 kg) and its acceleration (a ≈ 2306.82 m/s^2), we can calculate the magnitude of the net force using Newton's second law:
F = m * a
F = 0.058 kg * 2306.82 m/s^2
Calculating this, we get:
F ≈ 133.70 N
Therefore, the magnitude of the net force acting on the ball is approximately 133.70 Newtons.