A 7.00-kg bowling ball moves at 2.15 m/s. How fast must a 2.40-g Ping-Pong ball move so that the two balls have the same kinetic energy?
To find the speed at which the 2.40-g Ping-Pong ball must move to have the same kinetic energy as a 7.00-kg bowling ball, we can use the formula for kinetic energy:
Kinetic energy (KE) = 0.5 * mass * velocity^2
Let's denote the velocity of the Ping-Pong ball as v. As we want the kinetic energy of the Ping-Pong ball to be the same as that of the bowling ball, we can set up the equation:
0.5 * (mass of bowling ball) * (velocity of bowling ball)^2 = 0.5 * (mass of Ping-Pong ball) * v^2
Substituting the given values:
0.5 * 7.00 kg * (2.15 m/s)^2 = 0.5 * 0.00240 kg * v^2
Now, let's solve for v:
v^2 = (7.00 kg * (2.15 m/s)^2) / (0.00240 kg)
v^2 = 108.37 m^2/s^2 / 0.00240 kg
v^2 = 45154.17 m^2/s^2
v ≈ √(45154.17 m^2/s^2)
Using a calculator, we find that:
v ≈ 212.64 m/s
Therefore, the 2.40-g Ping-Pong ball must move at approximately 212.64 m/s to have the same kinetic energy as the 7.00-kg bowling ball.
To find out how fast the Ping-Pong ball must move, we need to equate the kinetic energies of both balls. The kinetic energy (KE) of an object is given by the equation KE = (1/2)mv^2, where m is the mass of the object and v is its velocity.
Given:
Mass of the bowling ball (m1) = 7.00 kg
Velocity of the bowling ball (v1) = 2.15 m/s
Mass of the Ping-Pong ball (m2) = 2.40 g = 0.00240 kg (convert grams to kilograms)
Now, we can set up an equation to equate the kinetic energies:
KE1 = KE2
Substituting the kinetic energy equation:
(1/2) (m1) (v1)^2 = (1/2) (m2) (v2)^2
Plugging in the values we know:
(1/2) (7.00 kg) (2.15 m/s)^2 = (1/2) (0.00240 kg) (v2)^2
Simplifying the equation:
51.2075 kg · m^2/s^2 = 6.48 × 10^-4 kg · m^2/s^2 · (v2)^2
Now, let's solve for (v2)^2:
(v2)^2 = (51.2075 kg · m^2/s^2) / (6.48 × 10^-4 kg · m^2/s^2)
(v2)^2 = 79,078.7 m^2/s^2
Taking the square root of both sides:
v2 = √79,078.7 m^2/s^2
Calculating the value for v2:
v2 ≈ 281.28 m/s
Therefore, the Ping-Pong ball must move at a velocity of approximately 281.28 m/s to have the same kinetic energy as the bowling ball.
Ke = .5 m v^2
.5 (7)(2.15)^2 = .5 (.0024) v^2
v^2 = (7)(2.15)^2/.0024