A 50 kg ice skater, initially at rest, throws a 0.15 kg snowball with a speed of 35 m/s. What is the approximate recoil speed of the skater?

Options

A. 0.10 m/s
B. 0.20 m/s
C. 0.70 m/s
D. 1.4 m/s

Explanation

To solve the problem of the ice skater throwing a snowball and determining the recoil speed of the skater, we can apply the principle of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Identify the masses and initial conditions
- Mass of the ice skater, \( M = 50 \, \text{kg} \)
- Mass of the snowball, \( m = 0.15 \, \text{kg} \)
- Initial speed of both the skater and the snowball, \( u = 0 \, \text{m/s} \) (since both are at rest)

### Step 2: Determine the final speed of the snowball
- Speed of the snowball after being thrown, \( v_s = 35 \, \text{m/s} \)

### Step 3: Apply the conservation of momentum
According to the law of conservation of momentum:
\[ \text{Initial momentum} = \text{Final momentum} \]
Since both the skater and the snowball are initially at rest, the initial momentum is:
\[ \text{Initial momentum} = 0 \]

The final momentum can be expressed as:
\[ \text{Final momentum} = m v_s + M v_k \]
where \( v_k \) is the recoil speed of the skater.

### Step 4: Set up the equation
Setting the initial momentum equal to the final momentum:
\[ 0 = m v_s + M v_k \]
Rearranging gives:
\[ M v_k = -m v_s \]

### Step 5: Solve for the recoil speed \( v_k \)
Substituting the known values:
\[ v_k = -\frac{m v_s}{M} \]
\[ v_k = -\frac{0.15 \, \text{kg} \times 35 \, \text{m/s}}{50 \, \text{kg}} \]

### Step 6: Calculate the value
Calculating the numerator:
\[ 0.15 \times 35 = 5.25 \, \text{kg m/s} \]
Now substituting into the equation:
\[ v_k = -\frac{5.25}{50} = -0.105 \, \text{m/s} \]

### Step 7: Interpret the result
The negative sign indicates that the direction of the skater's recoil speed is opposite to the direction of the snowball's speed. Therefore, the approximate recoil speed of the skater is:
\[ v_k \approx 0.105 \, \text{m/s} \]

### Final Answer
The approximate recoil speed of the skater is \( 0.10 \, \text{m/s} \) (in the opposite direction to the snowball).

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