A 58.0 kg pole vaulter running at 8.9 m/s vaults over the bar. If the vaulter's horizontal component of velocity over the bar is 1.0 m/s and air resistance is disregarded, how high was the jump?

To find the height of the jump, we can use the principles of projectile motion. The vaulter's vertical motion can be treated as a projectile, while the horizontal motion does not affect the height.

Let's start by finding the initial vertical velocity (Vy) of the vaulter. We know that the horizontal component of velocity (Vx) is 1.0 m/s, and the total velocity (V) is 8.9 m/s. We can use the Pythagorean theorem to find Vy:

V^2 = Vx^2 + Vy^2

Rearranging the equation:

Vy^2 = V^2 - Vx^2

Plugging in the given values:

Vy^2 = (8.9 m/s)^2 - (1.0 m/s)^2
Vy^2 = 79.21 m^2/s^2 - 1.0 m^2/s^2
Vy^2 = 78.21 m^2/s^2

Taking the square root of both sides:

Vy = √(78.21 m^2/s^2)
Vy ≈ 8.84 m/s (rounded to two decimal places)

Now that we have found the initial vertical velocity, we can use it to find the height of the jump. The height (h) can be determined using the equation:

h = (Vy^2) / (2 * g)

Where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Plugging in the values:

h = (8.84 m/s)^2 / (2 * 9.8 m/s^2)
h = 78.1456 m^2/s^2 / 19.6 m/s^2
h ≈ 3.99 m (rounded to two decimal places)

Therefore, the height of the jump is approximately 3.99 meters.

To determine the height of the jump, we can use the conservation of mechanical energy. The initial mechanical energy (kinetic energy) is equal to the final mechanical energy (potential energy), assuming no energy is lost due to air resistance.

The initial kinetic energy is given by the formula:

KE_initial = 0.5 * m * v_initial^2

where m is the mass of the vaulter (58.0 kg) and v_initial is the initial velocity (8.9 m/s).

KE_initial = 0.5 * 58.0 kg * (8.9 m/s)^2

KE_initial = 0.5 * 58.0 kg * 79.21 m^2/s^2

KE_initial = 2276.18 Joules

The final potential energy is given by the formula:

PE_final = m * g * h

where g is the acceleration due to gravity (9.8 m/s^2) and h is the height of the jump (unknown).

PE_final = 58.0 kg * 9.8 m/s^2 * h

PE_final = 568.4 h Joules

Since the initial kinetic energy is equal to the final potential energy, we can set them equal to each other:

KE_initial = PE_final

2276.18 Joules = 568.4 h Joules

Now, we can solve for h by rearranging the equation:

h = 2276.18 Joules / 568.4 Joules

h ≈ 4.00 meters

Therefore, the height of the jump is approximately 4.00 meters.