Answer :
Answer:
To find the input power of the cannon, we can use the formula for power:
Power = Work / Time
First, let's calculate the work done to lift the ball. The work done is equal to the force applied multiplied by the distance over which the force is applied. In this case, the force is equal to the weight of the ball and the distance is the height it is lifted.
Work = Force * Distance
The weight of the ball can be calculated using the formula:
Weight = Mass * Gravitational acceleration
where the mass of the ball is 1 kg and the acceleration due to gravity is approximately 9.8 m/s².
Weight = 1 kg * 9.8 m/s² = 9.8 N
The distance over which the force is applied is given as 10 m.
Now we can calculate the work done:
Work = 9.8 N * 10 m = 98 Joules
Since the cannon lifts 8 balls per minute, the time taken to lift one ball is 1/8 minutes or 60/8 seconds.
Time = 60/8 seconds = 7.5 seconds
Now we can calculate the power:
Power = Work / Time = 98 J / 7.5 s ≈ 13.07 Watts
However, this is the power for lifting one ball. Since the cannon lifts 8 balls per minute, we need to calculate the power for lifting all the balls in a minute.
Total Power = Power per ball * Number of balls per minute
Total Power = 13.07 W * 8 balls = 104.56 Watts
Finally, if the efficiency of the cannon is given as 80%, we can calculate the input power:
Input Power = Total Power / Efficiency
Input Power = 104.56 W / 0.80 = 130.7 Watts
Therefore, the input power of the cannon is approximately 130.7 Watts.
Answer: Given data:
The velocity of the ball when thrown: 72 km/h = 72 * 1000/3600 m/s = 20m/s
Height to which the ball is raised: 10 m
Number of balls lifted per minute: 8
Efficiency of the cannon: 80% = 0.8
Mass of each ball: 1 kg
Step-by-Step Calculation:
Calculate the Potential Energy of One Ball
Potential energy
�
�
E
p
of a ball raised to a height
ℎ
h is given by:
�
�
=
�
�
ℎ
E
p
=mgh
where:
�
m is the mass of the ball (1 kg),
�
g is the acceleration due to gravity (approximately
9.81
m/s
2
9.81 m/s
2
),
ℎ
h is the height the ball is raised (10 m).
�
�
=
1
×
9.81
×
10
=
98.1
joules
E
p
=1×9.81×10=98.1 joules
Step 2: Calculate the Work Done per Minute
Work done per minute
�
W to lift 8 balls is:
�
=
8
×
�
�
W=8×E
p
�
=
8
×
98.1
=
784.8
joules per minute
W=8×98.1=784.8 joules per minute
Step 3: Convert Work to Power
Power
�
P is defined as work done per unit time (in watts, where 1 watt = 1 joule/second):