A rock is kicked horizontally off a dock into the bay 3 times with a higher initial velocity each time as shown to the right. Which of the following statements is true regarding
these scenarios? AND WHY?

(a) The rock takes the longest time to hit the water in
case 3
(b) The rock takes about the same time to hit the water
in each case
(c) The rock hits the water below with the highest speed
in case 1
(d) The rock hits the water with the same speed in each|
case
Student Answer(s):
Student Reasoning:

The true statement is;

(b) The rock takes about the same time to hit the water in each case

Explanation:

The given information are;

The direction in which the rock is kicked = Horizontally

The number of different initial horizontal with which the rock as kicked =  different horizontal velocity

The time taken for the rock to hit the water is given by the following equation for projectile motion;

y = y₀ + v₀× sin(θ₀)×t - 1/2×g×t²

Where;

θ = The angle the rock makes with the horizontal when it was kicked = 0

y = The height through which the rock drops

y₀ = The initial height of the rock

t = The time taken for the ball to hit the water

g = The acceleration due to gravity ≈ 9.81 m/s²

For the different speeds, v₁, v₂, and v₃, we have;

y = y₀ + v₁× sin(0)×t - 1/2×g×t² = y₀ - 1/2×g×t²

y = y₀ + v₂× sin(0)×t - 1/2×g×t² = y₀ - 1/2×g×t²

y = y₀ + v₃× sin(0)×t - 1/2×g×t² = y₀ - 1/2×g×t²

Therefore, the time it takes for the ball to hit the water is given as follows;

Which is the same for each case

However, the speed, v, with which the rock hits the water is given by the following equation;

= The vertical velocity = v₀ × sin(θ₀) - g×t = v₀ × sin(0) - g×t = -g×t

= -g×t

Therefore, , is the same for all three times

While vₓ = v₁, v₂, and v₃, for each of the three times.

Therefore,  the speed with which the rock hits the water is different for all three times.

(a) The mass of the bullet is  and its speed is , so its momentum is


The pitcher claims that he can throw the ball with the same momentum p of the ball. Since the mass of the ball is m=0.148 kg, this means that the velocity of the ball must be:


(b) The kinetic energy of the bullet is:


while the kinetic energy of the ball is:


So, the bullet has greater kinetic energy than the ball.