A 2.42 kg steel ball strikes a massive wall at 11.7 m/s at an angle of 60.0° with the plane of the wall. It bounces off the wall with the same speed and angle, as shown in the figure below. If the ball is in contact with the wall for 0.118 s, what is the magnitude of the average force exerted by the wall on the ball?
To find the magnitude of the average force exerted by the wall on the ball, we can use the impulse-momentum principle.
The impulse-momentum principle states that the change in momentum of an object is equal to the impulse applied to it. In equation form, it can be expressed as:
Impulse = Change in Momentum
The impulse applied to the ball can be calculated using the following formula:
Impulse = Force × Time
The change in momentum of the ball can be calculated using the formula:
Change in Momentum = Mass × Change in Velocity
First, let's calculate the change in velocity of the ball. Since the ball bounces off the wall with the same speed and angle, the change in velocity is equal to twice the initial velocity.
Change in Velocity = 2 × Initial Velocity
The initial velocity can be calculated by decomposing the velocity into its x and y components:
Vx = Initial Velocity × cos θ
Vy = Initial Velocity × sin θ
where θ is the angle with the plane of the wall (60.0°) and the initial velocity is given as 11.7 m/s:
Vx = 11.7 m/s × cos(60.0°)
Vy = 11.7 m/s × sin(60.0°)
Next, we can calculate the change in velocity:
Change in Velocity = 2 × (Final Velocity - Initial Velocity)
The final velocity can be calculated using the same component values of the initial velocity:
Vx_final = 11.7 m/s × cos(60.0°)
Vy_final = -11.7 m/s × sin(60.0°)
Now, we can calculate the change in velocity:
Change in Velocity = 2 × [(Vx_final - Vx), (Vy_final - Vy)]
Next, we need to calculate the change in momentum of the ball:
Change in Momentum = Mass × Change in Velocity
The mass of the ball is given as 2.42 kg.
Finally, we can calculate the average force exerted by the wall on the ball using the impulse formula:
Average Force = Impulse / Time
Substituting the values calculated earlier:
Average Force = (Mass × Change in Velocity) / Time
Plug in the values to find the answer.
To find the magnitude of the average force exerted by the wall on the ball, we can use the impulse-momentum principle.
The impulse-momentum principle states that the change in momentum of an object is equal to the impulse applied to it.
The impulse can be calculated using the equation:
Impulse = Change in momentum
The change in momentum is given by:
Change in momentum = Final momentum - Initial momentum
The momentum of an object can be calculated using the equation:
Momentum = mass × velocity
Now, let's break down the problem step-by-step:
Step 1: Calculate the initial momentum of the ball.
Initial momentum = mass × initial velocity
Initial velocity = 11.7 m/s
Mass of the ball = 2.42 kg
Initial momentum = 2.42 kg × 11.7 m/s
Step 2: Calculate the final momentum of the ball.
Final momentum = mass × final velocity
Final velocity = 11.7 m/s (since the ball bounces off the wall with the same speed as before)
Final momentum = 2.42 kg × 11.7 m/s
Step 3: Calculate the change in momentum.
Change in momentum = Final momentum - Initial momentum
Step 4: Calculate the impulse applied to the ball.
Impulse = Change in momentum
Step 5: Calculate the average force exerted by the wall on the ball.
Average force = Impulse / Time
Time = 0.118 s
Putting it all together:
Initial momentum = 2.42 kg × 11.7 m/s
Final momentum = 2.42 kg × 11.7 m/s
Change in momentum = Final momentum - Initial momentum
Impulse = Change in momentum
Average force = Impulse / Time
Time = 0.118 s
By plugging in the values and performing the calculations, you can find the magnitude of the average force exerted by the wall on the ball.