An official major league baseball has a mass of 0.14 kg . A pitcher throws a 40
fastball which is hit by the batter straight back up the middle at a speed of
.
What is the impulse of the ball during the collision with the bat?
Woint
Numeric 1 point
If this collision occurs during a time of 0.012 seconds, what is the av
force exerted by the bat on the ball?
Type your answer.

Ask by Stanley Whittaker. in the United States

Jan 25,2025

Question

An official major league baseball has a mass of 0.14 kg . A pitcher throws a 40
fastball which is hit by the batter straight back up the middle at a speed of
.
What is the impulse of the ball during the collision with the bat?
Woint
Numeric 1 point
If this collision occurs during a time of 0.012 seconds, what is the av
force exerted by the bat on the ball?
Type your answer.

Ask by Stanley Whittaker. in the United States

Jan 25,2025

UpStudy AI · Accepted Answer

To solve this problem, we will follow these steps:

1. **Calculate the impulse of the ball during the collision with the bat.**
2. **Calculate the average force exerted by the bat on the ball.**

### Step 1: Calculate the Impulse

Impulse ($$ J $$) is defined as the change in momentum of an object. It can be calculated using the formula:

$$
J = \Delta p = m(v_f - v_i)
$$

Where:
- $$ m $$ is the mass of the ball (0.14 kg),
- $$ v_f $$ is the final velocity of the ball after being hit (46 m/s),
- $$ v_i $$ is the initial velocity of the ball before being hit (-40 m/s, negative because it is in the opposite direction).

Now, we can substitute the values into the formula:

$$
J = 0.14 \, \text{kg} \times (46 \, \text{m/s} - (-40 \, \text{m/s}))
$$

Calculating the change in velocity:

$$
v_f - v_i = 46 + 40 = 86 \, \text{m/s}
$$

Now substituting back into the impulse formula:

$$
J = 0.14 \, \text{kg} \times 86 \, \text{m/s}
$$

Now, let's calculate this value.
Calculate the value by following steps:
- step0: Calculate:
  $$0.14\times 86$$
- step1: Multiply the numbers:
  $$12.04$$
The impulse of the ball during the collision with the bat is:

$$
J = 12.04 \, \text{kg m/s}
$$

### Step 2: Calculate the Average Force

The average force ($$ F $$) exerted by the bat on the ball can be calculated using the formula:

$$
F = \frac{J}{\Delta t}
$$

Where:
- $$ J $$ is the impulse (12.04 kg m/s),
- $$ \Delta t $$ is the time duration of the collision (0.012 seconds).

Now we can substitute the values into the formula:

$$
F = \frac{12.04 \, \text{kg m/s}}{0.012 \, \text{s}}
$$

Let's calculate this value.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{12.04}{0.012}$$
- step1: Convert the expressions:
  $$\frac{\frac{301}{25}}{\frac{3}{250}}$$
- step2: Multiply by the reciprocal:
  $$\frac{301}{25}\times \frac{250}{3}$$
- step3: Reduce the numbers:
  $$301\times \frac{10}{3}$$
- step4: Multiply:
  $$\frac{301\times 10}{3}$$
- step5: Multiply:
  $$\frac{3010}{3}$$
The average force exerted by the bat on the ball is:

$$
F \approx 1003.33 \, \text{N}
$$

### Summary of Results:
- **Impulse of the ball:** $$ 12.04 \, \text{kg m/s} $$
- **Average force exerted by the bat on the ball:** $$ 1003.33 \, \text{N} $$
The impulse of the ball during the collision is 12.04 kg·m/s, and the average force exerted by the bat on the ball is approximately 1003.33 N.

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