Question

A scientist determines that an alpha particle has an initial kinetic energy of 8.0 x 10⁻¹⁴ J. The mass of an alpha particle is known to be 6.67 x 10²⁷ kg. Solve for the velocity of the alpha particle. What is the speed when expressed as a fraction of the speed of light?

Answer 1

The velocity of an alpha particle with an initial kinetic energy of 8.0 x 10⁻¹⁴ J is approximately 1.54 x 10⁶ m/s, which is about 0.513% the speed of light.

Step-by-step explanation:

To solve for the velocity of the alpha particle, we can use the relationship between kinetic energy (KE) and velocity (v), given by the equation KE = ½ mv², where m is the mass of the particle. Plugging in the given values, we get:

8.0 x 10⁻¹⁴ J = ½ (6.67 x 10²⁷ kg) v²

Solving for v, we obtain:

v = sqrt((2 * 8.0 x 10⁻¹⁴ J) / (6.67 x 10²⁷ kg))

After calculating, the velocity v is:

v ≈ 1.54 x 10⁶ m/s

The speed of light, c, is approximately 3.00 x 10⁸ m/s. To express the speed of the alpha particle as a fraction of the speed of light, we divide the velocity of the alpha particle by the speed of light:

(1.54 x 10⁶ m/s) / (3.00 x 10⁸ m/s) ≈ 5.13 x 10⁻³

So, the speed of the alpha particle is approximately 0.513% the speed of light.