Draw the graph
On its way along a straight line, a car starts from rest and accelerated at a rate of 2.5 m/s² for 10s. It maintained its speed for further 5 s and decelerated by 0.5m/s² to come to rest. What is the time taken for the entire motion? How far did it travel? Draw the position vs time, and velocity vs time graph of the motion of the car
To calculate the time taken for the entire motion and the distance traveled, we need to analyze each segment of the car's motion separately.
1. Acceleration phase (0-10 seconds):
During this phase, the car starts from rest and accelerates at a rate of 2.5 m/s² for 10 seconds. To find the distance traveled during this phase, we can use the formula:
distance = (initial velocity * time) + (0.5 * acceleration * time²)
Given:
Initial velocity (u) = 0 m/s
Acceleration (a) = 2.5 m/s²
Time (t) = 10 s
distance = (0 * 10) + (0.5 * 2.5 * 10²)
distance = 0 + (0.5 * 2.5 * 100)
distance = 0 + 0.5 * 250
distance = 125 meters
2. Constant velocity phase (10-15 seconds):
During this phase, the car maintains a constant velocity for 5 seconds. Since velocity is constant, the distance traveled can be calculated as:
distance = velocity * time
Given:
velocity = final velocity from previous phase = 2.5 m/s
time = 5 s
distance = 2.5 * 5
distance = 12.5 meters
3. Deceleration phase (15-20 seconds):
During this phase, the car decelerates at a rate of 0.5 m/s² until it comes to rest. To find the time taken and distance traveled during this phase, we can use the formulas:
final velocity = initial velocity - (acceleration * time)
distance = (initial velocity * time) - (0.5 * acceleration * time²)
Given:
Initial velocity = 2.5 m/s
Acceleration = -0.5 m/s² (negative sign indicates deceleration)
Final velocity = 0 m/s
To find the time taken, we can rearrange the formula for final velocity:
0 = 2.5 - (0.5 * time)
0.5 * time = 2.5
time = 2.5 / 0.5
time = 5 seconds
Now, we can find the distance:
distance = (2.5 * 5) - (0.5 * (-0.5) * 5²)
distance = 12.5 - (0.5 * 0.25 * 25)
distance = 12.5 - 3.125
distance = 9.375 meters
Therefore, the time taken for the entire motion is 10 seconds + 5 seconds + 5 seconds = 20 seconds.
The total distance traveled is 125 meters + 12.5 meters + 9.375 meters = 146.875 meters.
To draw the position vs time graph, we can plot the distance traveled at various time intervals shown above:
0s: 0m
10s: 125m
15s: 137.5m (125m + 12.5m)
20s: 146.875m (125m + 12.5m + 9.375m)
The graph will show a linear increase in distance for the first 10 seconds, followed by a flat line segment from 10 to 15 seconds, and then a decreasing slope from 15 to 20 seconds.
To draw the velocity vs time graph, we need to calculate the velocities at various time intervals:
0-10s: Constant acceleration of 2.5 m/s²
Velocity after 10 seconds:
v = u + (a * t) = 0 + (2.5 * 10) = 25 m/s
10-15s: Constant velocity of 25 m/s
15-20s: Deceleration of -0.5 m/s²
Velocity at 20 seconds:
v = u + (a * t) = 25 + (-0.5 * 5) = 22.5 m/s
The graph will show an increasing slope from 0 to 10 seconds, a flat line segment at 25 m/s from 10 to 15 seconds, and a decreasing slope from 15 to 20 seconds.