While sitting on a frozen pond, a 35kg child throws a 0.5kg ball horizontally at a velocity of +6m/s. Calculate the resulting velocity of the child across the ice.Suppose the child throws the ball upward instead, at a 25o angle above horizontal. a) Calculate the resulting velocity of the child across the ice. b) Calculate the vertical impulse delivered to the child (and therefore, to the ice).

To calculate the resulting velocity of the child across the ice in both scenarios, we need to apply the principle of conservation of momentum.

a) When the child throws a ball horizontally, no vertical momentum is involved. Thus, the horizontal momentum is conserved. The initial momentum of the system is given by the product of the mass of the child and their initial velocity, which is (35 kg) × (0 m/s) = 0 kg·m/s. The final momentum is the sum of the momenta of the child and the ball.

Let's assume the velocity of the child across the ice after throwing the ball is Vc, and the velocity of the ball is Vb. Since the ball was thrown horizontally, Vb is 6 m/s in the positive x-direction. Thus, the final momentum can be calculated as the sum of the momentum of the child and the momentum of the ball:

Final momentum = (mass of the child × velocity of the child) + (mass of the ball × velocity of the ball)
Final momentum = (35 kg × Vc) + (0.5 kg × 6 m/s)

Since momentum is conserved, the initial and final momentum must be equal:

0 kg·m/s = (35 kg × Vc) + (0.5 kg × 6 m/s)

Simplifying the equation:

0 = 35Vc + 3

Rearranging the equation to solve for Vc:

35Vc = -3

Vc = -3 / 35

The resulting velocity of the child across the ice when the ball is thrown horizontally is approximately -0.086 m/s. Note that the negative sign indicates that the child moves in the opposite direction of the initial velocity of the ball.

b) When the child throws the ball upward at a 25° angle above horizontal, vertical momentum is involved. To calculate the vertical impulse delivered to the child, we need to consider the change in vertical momentum.

The initial vertical momentum is given by the product of the mass of the child and their initial vertical velocity, which is zero because the child is sitting on the frozen pond. The final vertical momentum is the sum of the vertical momenta of the child and the ball.

Let's assume the vertical velocity of the child after throwing the ball is Vcy, and the vertical velocity of the ball is Vby. The final vertical momentum can be calculated as:

Final vertical momentum = (mass of the child × vertical velocity of the child) + (mass of the ball × vertical velocity of the ball)
Final vertical momentum = (35 kg × Vcy) + (0.5 kg × Vby)

Since momentum is conserved, the initial and final vertical momentum must be equal:

0 kg·m/s = (35 kg × 0) + (0.5 kg × Vby)

Simplifying the equation:

0 = 0 + 0.5Vby

Vby = 0

The vertical velocity of the ball after throwing is zero because it only travels horizontally. Therefore, the final vertical momentum is zero.

The change in vertical momentum is equal to the final vertical momentum minus the initial vertical momentum:

Change in vertical momentum = Final vertical momentum - Initial vertical momentum
Change in vertical momentum = 0 - 0
Change in vertical momentum = 0

The vertical impulse delivered to the child (and therefore to the ice) is zero.