A 0.165-kg ball, moving in the positive direction at 10 m/s, is acted on by the impulse shown in the graph below. What is the ball's speed at 4.0 s?
Determine the momentum of 750 Kg car moving westward at 10 m/s.
To find the ball's speed at 4.0 s, we need to determine the change in momentum of the ball when it is acted upon by the impulse. This can be done by calculating the area under the impulse-time graph.
First, let's analyze the impulse-time graph provided and determine the area corresponding to the impulse:
- The impulse is represented by the area under the graph, which in this case is a triangle.
- The base of the triangle is given by the time interval of 4.0 s.
- The height of the triangle represents the force acting on the ball during this time.
Since the impulse is equal to the change in momentum, we can calculate it as follows:
Impulse = change in momentum = mass × change in velocity
Given:
Mass of the ball (m) = 0.165 kg
Initial velocity of the ball (u) = 10 m/s
Time interval (t) = 4.0 s
From the graph, we can determine the height (h) of the triangle:
Height (h) = 50 N
Now, we can calculate the change in momentum using the formula:
Impulse = mass × change in velocity
Impulse = (0.165 kg) × change in velocity
To find the change in velocity, we need to calculate the velocity at 4.0 s. To do this, we can use the equation of motion:
Final velocity (v) = Initial velocity (u) + acceleration (a) × time (t)
Since the acceleration is constant (constant deceleration due to the force), we can calculate it first using the formula:
Force (F) = mass (m) × acceleration (a)
Given:
Force (F) = Height of the triangle (h) = 50 N
Mass (m) = 0.165 kg
Using the formula:
a = F / m
Acceleration (a) = (50 N) / (0.165 kg)
Now, we can calculate the final velocity (v) at 4.0 s using the equation of motion:
v = u + a × t
v = 10 m/s + (acceleration) × 4.0 s
Once we have the final velocity (v) at 4.0 s, we can calculate the change in velocity and subsequently find the impulse:
change in velocity = v - u
Finally, we can calculate the impulse by substituting the mass (m) and change in velocity into the equation:
Impulse = (0.165 kg) × (change in velocity)
From the impulse, we can then determine the ball's speed at 4.0 s by dividing the impulse by the mass of the ball and adding it to the initial velocity:
Speed at 4.0 s = (Impulse / m) + u
To find the ball's speed at 4.0 seconds, we need to calculate the impulse applied to the ball and use it to determine the change in velocity.
Impulse is defined as the product of the force applied to an object and the time interval over which the force is applied. Mathematically, impulse can be calculated using the formula:
Impulse = Change in momentum
In this case, the impulse graph is given, so we need to calculate the area under the graph to find the total impulse applied to the ball.
To find the impulse applied to the ball, we need to determine the area of the graph. The graph represents force (F) as a function of time (t). The impulse is equal to the area under the graph.
By calculating the area under the graph, we can find the impulse value. Then, using the impulse-momentum theorem, we can determine the change in momentum. Since the ball's mass is given as 0.165 kg, we can calculate the change in velocity.
Once we find the change in velocity, we can use it to find the ball's speed at 4.0 seconds.
To calculate the area under the graph, we can use numerical integration techniques such as the trapezoidal rule or Simpson's rule. These methods approximate the area under the curve by dividing it into smaller trapezoids or segments, respectively.
Here's how you can use the trapezoidal rule to approximate the area under the graph:
1. Divide the time interval into small equal intervals. In this case, we need to divide the interval from 0 to 4 seconds into smaller intervals.
For example, you can choose Δt = 0.1 seconds, which means you'll have 40 sub-intervals in total.
2. Calculate the force at each sub-interval by reading the values from the graph.
3. Multiply each force value by the time interval Δt to obtain the area of each smaller trapezoid.
4. Sum up all the areas to get the total impulse applied to the ball.
Once you have the impulse, you can use the impulse-momentum theorem:
Impulse = Change in momentum = mass * (final velocity - initial velocity)
Using this equation, you can rearrange it to solve for the final velocity:
final velocity = (impulse / mass) + initial velocity.
Plug in the values you obtained to calculate the ball's speed at 4.0 seconds.