When serving a tennis ball, a player hits the ball when its velocity is zero (at the highest point of a vertical toss). The racquet exerts a force of 525 N on the ball for 4.8 ms, giving it a final velocity of 47.5 m/s.
Using the data, find the mass of the ball, in grams.
Numeric : A numeric value is expected and not an expression.
m =
The correct answer and explanation is :
To solve for the mass of the tennis ball, we can use the principles of impulse and momentum. Impulse is the change in momentum, and it is equal to the force applied to an object multiplied by the time over which the force is applied.
Step 1: Define the given values
- Force (F) = 525 N
- Time (Δt) = 4.8 ms = 4.8 × 10⁻³ s
- Final velocity (v) = 47.5 m/s
- The initial velocity (u) = 0 m/s (since the ball starts at rest when it reaches the highest point of the toss).
Step 2: Use the impulse-momentum theorem
The impulse-momentum theorem states that the impulse on an object is equal to the change in its momentum. Mathematically, this is written as:
[
\text{Impulse} = \Delta p = F \cdot \Delta t
]
Since momentum ( p ) is given by ( p = m \cdot v ), where ( m ) is the mass and ( v ) is the velocity, the change in momentum ( \Delta p ) is:
[
\Delta p = m \cdot (v – u)
]
Since the initial velocity ( u = 0 ), this simplifies to:
[
\Delta p = m \cdot v
]
Step 3: Relate impulse to momentum change
From the impulse-momentum theorem, we have:
[
F \cdot \Delta t = m \cdot v
]
Now, solve for the mass ( m ):
[
m = \frac{F \cdot \Delta t}{v}
]
Step 4: Plug in the known values
[
m = \frac{525 \, \text{N} \cdot 4.8 \times 10^{-3} \, \text{s}}{47.5 \, \text{m/s}}
]
[
m = \frac{2.52 \, \text{Ns}}{47.5 \, \text{m/s}}
]
[
m \approx 0.053 \, \text{kg}
]
Step 5: Convert mass to grams
Since 1 kg = 1000 grams, we convert the mass from kilograms to grams:
[
m = 0.053 \, \text{kg} \times 1000 = 53 \, \text{grams}
]
Final Answer:
The mass of the ball is 53 grams.
Explanation:
The impulse applied to the tennis ball is calculated by multiplying the force exerted on the ball by the time over which the force acts. This impulse causes a change in momentum, which is related to the mass and final velocity of the ball. By rearranging the impulse-momentum relationship and solving for the mass, we find that the mass of the tennis ball is 53 grams. This calculation involves basic concepts from mechanics, such as impulse, momentum, and the relationship between force, time, and velocity.