The figure below shows the acceleration as a function of time for an object.
(a) If the object starts from rest at t = 0, what is the velocity of the object as a function of time?
for 0s < t < 10s v(t) = m/s
for 10s < t < 26s v(t) = m/s
(b) If the object instead has a velocity of −40 m/s at t = 0, how does your result for part (a) change?
for 0 < t < 10s v(t) = m/s
for 10s < t < 26s v(t) = m/s
you will be unable to post figures here.
To determine the velocity of the object as a function of time, given the acceleration-time graph, we need to integrate the acceleration graph with respect to time.
(a) If the object starts from rest at t=0, we need to find the area under the acceleration-time graph to find the change in velocity. The change in velocity over each time interval will give us the velocity function.
For the time interval 0s < t < 10s, the acceleration is constant and equal to a m/s². The area under this portion of the graph is given by:
Area = a * (10s - 0s) = a * 10s
This area represents the change in velocity over this time interval, so the velocity at any time t within this interval is:
v(t) = a * t
where 'a' is the constant acceleration.
For the time interval 10s < t < 26s, the acceleration remains constant at a different value (let's call it b m/s²). The area under this portion of the graph is given by:
Area = b * (26s - 10s) = b * 16s
This area represents the change in velocity over this time interval, so the velocity at any time t within this interval is:
v(t) = v(10s) + b * (t - 10s)
Here, v(10s) represents the velocity at t=10s, which can be found using the previous calculation.
(b) If the object instead has a velocity of -40 m/s at t=0, the initial conditions change. We will need to incorporate the initial velocity into our calculations.
For the time interval 0s < t < 10s, the calculation remains the same as before since the initial velocity does not affect the area under the acceleration-time graph.
For the time interval 10s < t < 26s, we need to find the new initial velocity at t=10s. Since the object has an initial velocity of -40 m/s, the velocity function becomes:
v(t) = -40 + b * (t - 10s)
Here, the initial condition of -40 m/s at t=0 is taken into account by subtracting it from the velocity function.
So, to summarize:
(a) If the object starts from rest at t = 0:
- for 0s < t < 10s: v(t) = a * t
- for 10s < t < 26s: v(t) = v(10s) + b * (t - 10s)
(b) If the object has an initial velocity of -40 m/s at t = 0:
- for 0s < t < 10s: v(t) = -40 + a * t
- for 10s < t < 26s: v(t) = -40 + b * (t - 10s)