How much work does it take to accelerate a 15-kg ball from rest to 5 m/s? What is the power generated?
The correct answer and explanation is:
To determine the work and power generated in accelerating the 15-kg ball from rest to 5 m/s, we can follow these steps:
Work Done (W):
Work is defined as the change in kinetic energy, which can be calculated using the formula for kinetic energy: KE=12mv2KE = \frac{1}{2} m v^2
Where:
- mm is the mass of the object (15 kg)
- vv is the final velocity (5 m/s)
- The initial velocity is 0 since the ball starts from rest.
So, the work done to accelerate the ball is the kinetic energy it gains: W=ΔKE=12mv2=12×15 kg×(5 m/s)2W = \Delta KE = \frac{1}{2} m v^2 = \frac{1}{2} \times 15 \, \text{kg} \times (5 \, \text{m/s})^2 W=12×15×25=187.5 JW = \frac{1}{2} \times 15 \times 25 = 187.5 \, \text{J}
Therefore, the work done to accelerate the ball is 187.5 joules.
Power Generated (P):
Power is the rate at which work is done, calculated using the formula: P=WtP = \frac{W}{t}
Where:
- PP is the power
- WW is the work done
- tt is the time taken to do the work
To calculate the power generated, we need to know the time taken to accelerate the ball. If this time is not given, power cannot be calculated directly. However, if we assume the time t=2 secondst = 2 \, \text{seconds} (as an example), the power generated would be: P=187.5 J2 s=93.75 WP = \frac{187.5 \, \text{J}}{2 \, \text{s}} = 93.75 \, \text{W}
Thus, with a time of 2 seconds, the power generated would be 93.75 watts.
Conclusion:
- The work required to accelerate the 15-kg ball from rest to 5 m/s is 187.5 joules.
- The power generated depends on the time taken, and with an assumed time of 2 seconds, the power would be 93.75 watts.