A 1.5-kg playground ball is moving with a velocity of 3.0 m/s directed 30
°
below thehorizontal just before it strikes a horizontal surface. The ball leaves this surface 0.50 slater with a velocity of 2.0 m/s directed 60
°
above the horizontal. What is themagnitude of the average resultant force on the ball?
Divide the magnitude of the momentum changr by 0.50 s. You will need to compute both the horizontal and vertical components of the momentum change.
Well, it seems like this ball had quite the adventure. Let's calculate the magnitude of the average resultant force, shall we?
To find the average resultant force, we can use Newton's second law, which states that force (F) is equal to mass (m) times acceleration (a). In this case, we'll have to calculate the change in velocity (Δv) over the time interval (Δt), and divide it by the mass of the ball (m).
Δv = 2.0 m/s - 3.0 m/s = -1.0 m/s
Now, let's determine the horizontal and vertical components of the Δv:
Δv_horizontal = Δv * cos(60°)
Δv_vertical = Δv * sin(60°)
Δv_horizontal = -1.0 m/s * cos(60°) = -1.0 m/s * 0.5 = -0.5 m/s
Δv_vertical = -1.0 m/s * sin(60°) = -1.0 m/s * √3/2 ≈ -0.87 m/s
Since we know the change in velocity, we can now calculate the average acceleration:
a_horizontal = Δv_horizontal / Δt = -0.5 m/s / 0.5 s = -1.0 m/s²
a_vertical = Δv_vertical / Δt = -0.87 m/s / 0.5 s = -1.74 m/s²
Now, let's find the average resultant acceleration using Pythagoras:
a_resultant = √(a_horizontal² + a_vertical²) = √((-1.0 m/s²)² + (-1.74 m/s²)²) ≈ √(2.5 m/s²) ≈ 1.58 m/s²
Finally, we can find the average resultant force by multiplying the mass of the ball with the average resultant acceleration:
F_resultant = m * a_resultant = 1.5 kg * 1.58 m/s² = 2.37 N
So, the magnitude of the average resultant force on the ball is approximately 2.37 Newtons. And just like that, our ball goes from playground hero to physics superstar!
To find the magnitude of the average resultant force on the ball, we can use the equation:
Average resultant force = (change in momentum) / (change in time)
1. First, let's find the change in momentum. The momentum of an object can be calculated by multiplying its mass by its velocity.
Initial momentum = mass * initial velocity
= 1.5 kg * 3.0 m/s
Final momentum = mass * final velocity
= 1.5 kg * 2.0 m/s
Change in momentum = Final momentum - Initial momentum
2. Next, let's find the change in time. The given information states that the ball leaves the surface 0.50 s later. Therefore, the change in time is 0.50 s.
3. Now, we can substitute the values into the equation:
Average resultant force = (change in momentum) / (change in time)
= (Final momentum - Initial momentum) / (0.50 s)
4. Calculate the final result:
Average resultant force = [1.5 kg * 2.0 m/s - 1.5 kg * 3.0 m/s] / 0.50 s
Simplify the equation:
Average resultant force = [-1.5 kg * 1.0 m/s] / 0.50 s
Average resultant force = -3.0 N
The magnitude of the average resultant force on the ball is 3.0 N.
To find the magnitude of the average resultant force on the ball, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
1. Let's first find the initial momentum of the ball before the collision:
Initial momentum along x-axis (horizontal component) = mass × velocity × cos(angle)
= 1.5 kg × 3.0 m/s × cos(30°)
2. Next, let's find the initial momentum of the ball before the collision along the y-axis (vertical component):
Initial momentum along y-axis (vertical component) = mass × velocity × sin(angle)
= 1.5 kg × 3.0 m/s × sin(30°)
3. Now, let's find the final momentum of the ball after the collision along the x-axis:
Final momentum along x-axis = mass × velocity × cos(angle)
= 1.5 kg × 2.0 m/s × cos(60°)
4. Finally, let's find the final momentum of the ball after the collision along the y-axis:
Final momentum along y-axis = mass × velocity × sin(angle)
= 1.5 kg × 2.0 m/s × sin(60°)
5. To calculate the magnitude of the average resultant force, we need to find the change in momentum in both the x and y directions and then calculate the magnitude:
Change in momentum along x-axis = Final momentum along x-axis - Initial momentum along x-axis
Change in momentum along y-axis = Final momentum along y-axis - Initial momentum along y-axis
Magnitude of the average resultant force = Square root[(Change in momentum along x-axis)^2 + (Change in momentum along y-axis)^2]
By substituting the values obtained in steps 1-4, you can calculate the magnitude of the average resultant force on the ball.