A remote-controlled toy car of mass 3.3 kg starts from rest at the origin at t = 0 and moves in the positive direction of an x axis. The net force on the car as a function of time is given by the figure. (a) What is the time rate of change of the momentum of the car at t = 3.0 s? (b) What is the momentum of the car at t = 3.0 s?
To find the time rate of change of momentum of the car at t = 3.0 s, we need to calculate the derivative of the momentum with respect to time at that specific time.
The momentum of an object is defined as the product of its mass and velocity. In mathematical terms, momentum (p) is given by:
p = m * v
where m is the mass and v is the velocity.
In this problem, the mass of the car is given as 3.3 kg. However, we don't have any information about the velocity of the car. But we do know that the net force acting on the car is given as a function of time.
To relate force, mass, and acceleration, we can use Newton's second law of motion:
F = m * a
where F is the net force, m is the mass, and a is the acceleration.
We know that acceleration (a) is the second derivative of position (x) with respect to time (t):
a = d²x / dt²
From the figure, we see that the net force on the car is constant until t = 2.5 s, and then it decreases linearly until t = 4.0 s.
Let's break down the problem into two parts:
1. From t = 0 to t = 2.5 s:
- The net force is constant, let's call it F1.
- Using Newton's second law, we can find the acceleration during this time interval: a1 = F1 / m.
- We can integrate the acceleration to find the velocity: v1 = ∫ a1 dt.
- Then integrate the velocity to find the position: x1 = ∫ v1 dt.
- Note that at t = 2.5 s, the velocity and position will be increasing from zero.
2. From t = 2.5 s to t = 4.0 s:
- The net force is linearly decreasing as given by the figure.
- We can calculate the acceleration at any given time within this interval.
- We can then integrate the acceleration to find the velocity: v2 = ∫ a2 dt.
- Similarly, we integrate the velocity to find the position: x2 = ∫ v2 dt.
- Note that at t = 4.0 s, the velocity and position will be decreasing, and finally, the car will come to a stop.
To find the time rate of change of momentum at t = 3.0 s, we need to consider both intervals and calculate the derivative of the momentum function with respect to time at that specific time.