I'm having trouble with motion graphs.
How do you go from an acceleration-time graph to a velocity-time graph or displacement time graph?
I know that to go from v-t to a-t I just take the slope of the v-t graph, but how do you do it the other way around?
To go from an acceleration-time graph to a velocity-time graph or displacement-time graph, you need to integrate the acceleration-time graph.
First, let's start with going from an acceleration-time (a-t) graph to a velocity-time (v-t) graph:
1. Identify the area between the acceleration-time graph and the time axis. This area represents the change in velocity.
2. Integrate the acceleration-time graph to find the velocity-time graph. Integration involves finding the cumulative area under the curve.
3. To integrate, start by identifying the initial velocity at a given time, usually t = 0. This is done by locating the point on the velocity-time axis that corresponds to t = 0.
4. Next, integrate the area under the acceleration-time graph by summing the areas of each small rectangle or trapezoid between the graph and the time axis. The area can be calculated using the formula: area = width × height.
5. Add the areas of each rectangle or trapezoid to find the total change in velocity over a given time interval.
6. To obtain the velocity-time graph, plot the cumulative change in velocity (from integrating) for each corresponding time interval.
Now, let's move on to going from an acceleration-time (a-t) graph to a displacement-time (d-t) graph:
1. Identify the area between the acceleration-time graph and the time axis. This area represents the change in velocity.
2. Integrate the acceleration-time graph to find the velocity-time graph as explained previously.
3. Next, repeat the integration process on the velocity-time graph to find the displacement-time graph. This involves finding the cumulative area under the velocity-time graph.
4. Again, start with the initial displacement at a given time, usually t = 0.
5. Integrate the area under the velocity-time graph by summing the areas of each small rectangle or trapezoid between the graph and the time axis.
6. Add the areas of each rectangle or trapezoid to find the total change in displacement over a given time interval.
7. To obtain the displacement-time graph, plot the cumulative change in displacement (from integrating) for each corresponding time interval.
Remember that integration involves finding the cumulative area under the graph, and this process is done to go from an acceleration-time graph to a velocity-time graph and then to a displacement-time graph.
To go from an acceleration-time (a-t) graph to a velocity-time (v-t) graph or displacement-time (d-t) graph, you need to integrate the acceleration values.
Step 1: Start with the a-t graph. The a-t graph represents the rate of change of velocity.
Step 2: Integrate the acceleration values. This means you need to find the area under the a-t graph to obtain the change in velocity (Δv). This can be done by calculating the definite integral of the acceleration function.
For example, if you have a constant acceleration of 5 m/s² for a duration of 10 seconds, then the change in velocity would be:
Δv = ∫(5) dt = 5t + C = 5(10) + C = 50 + C, where C is the constant of integration.
Step 3: Determine the initial velocity (v₀) value. To determine v₀, you need to know the initial conditions of the motion. If the initial velocity is known, you can add it to the change in velocity (Δv).
For example, if the initial velocity (v₀) is 15 m/s, then the velocity-time (v-t) graph would be:
v = 15 + Δv = 15 + (50 + C) = 65 + C.
Step 4: Obtain the displacement-time (d-t) graph using the velocity-time (v-t) graph. To do this, you need to integrate the velocity function.
Integrate the v-t graph by finding the area under the curve to determine the change in position or displacement (Δd). This can be done by calculating the definite integral of the velocity function.
For example, if you have a velocity function of v = 65 + C, integrate it to find the displacement:
Δd = ∫(65 + C) dt = 65t + Ct + D = 65(10) + C(10) + D = 650 + 10C + D, where D is the constant of integration.
Step 5: Determine the initial position (d₀) value. The initial position (d₀) is required to get the displacement-time (d-t) graph. If the initial position is known, you can add it to the change in position (Δd).
For example, if the initial position (d₀) is 200 m, then the displacement-time (d-t) graph would be:
d = 200 + Δd = 200 + (650 + 10C + D) = 850 + 10C + D.
By following these steps and integrating the respective graphs, you can go from an acceleration-time (a-t) graph to a velocity-time (v-t) graph or displacement-time (d-t) graph.