If an object has a constant velocity, then its acceleration (which is defined as the rate of change of velocity) is zero. Because the velocity is contant, the velocity at any instant of time is the same as the average velocity calculated during any interval of time. Let's look at an example of a golf ball that moves with a contant velocity in the animation below. Graphs of the x-position and y-position as a function of time are shown.
What is the velocity of the ball? Hmm, since the ball's position remains constant and velocity is the change in position divided by a time interval, then the velocity of the ball is zero! That's not interesting, but is still an example of constant velocity. Check out the graphs (note that the data is plotted on the graph in real-time). At every instant of time, the position of the ball is the same; therefore, the graphs of x vs. t and of y vs. t are just flat lines.
Let's look at another example where the golf ball moves with a constant velocity in the +x direction.
Since the ball does not move in the y-direction at all, what is its y-velocity? It's zero because the ball's y-position never changes. That's why the graph of y vs. t is a flat line.
The x-velocity, however, is positive and constant. How do we know? Look at the slope of the x vs. t graph. The x-velocity is the slope of the line on a graph of x vs. t. Since the function is a line of constant slope (i.e. a straight line), the x-velocity is constant. Measure the value of the slope, and you have the x-velocity. In this example, the slope of the x vs. t graph is 2.0 m/s, so vx = 2.0 m/s
The mathematical equation that describes the relationship between x and t on a graph of constant velocity is
where x0 is the x-position at t=0. If you want to relate velocity to any displacement during a certain time interval, just use the definition of average velocity
1. The animation below shows a golf ball that moves with a constant velocity in a straight line, but not along one of the axes (horizontal or vertical).
(a) What is the x-velocity of the golf ball? vx = <_> m/s
(b) What is the y-velocity of the golf ball? vy = <_> m/s
(c) What is the speed of the golf ball?
2. Suppose that your reaction time is 1.2 s. If your car is traveling at a constant speed of 30 m/s and the car ahead of you slams on the brakes, how far will your car travel before you can hit the brakes? (Because the choice of coordinate system does not affect the answer, assume for the sake of simplicity that you are traveling in the +x direction.)
Technically, the ball moving with constant velocity also had a constant acceleration--zero acceleration. However, let's consider constant, non-zero acceleration. This is a more difficult concept to grasp than contant velocity. So practice, practice, practice thinking about it.
Here's an example of the golf ball moving linearly (i.e. along a straight line) with constant acceleration in the +x direction.
First, note that the object was speeding up. That is, the magnitude of the x-velocity increased with time. You can notice this by looking at the graph. The slope of a line tangent to the graph of x vs. t is the instantaneous x-velocity. Since the function of x vs. t gets steeper and steeper, the ball is speeding up. Also, notice that the slope of a line tangent to the curve is always positive; therefore, vx is always positive. That makes sense because the ball is always moving to the right.
Now, let's look at an animation where the ball is initially moving to the right but has a x-acceleration that is negative (i.e. to the left). This is like being in a moving car and hitting the breaks. Except in a car, when it stops the acceleration becomes zero (that's why you remain stopped). However, in the case of this golf ball, the acceleration remains constant and in the -x direction. It never goes to zero. Check out what happens!
Examine the x vs. t graph. The slope of a tangent line on the graph is initially positive. It then gets smaller, goes to zero (at the peak on the graph), and then becomes negative and gets steeper. The slope of the tangent line tells you what is happening to the x-velocity. It is initially large and positive, but decreases to zero (slows down), becomes negative, and increases (speeds up).
By the way, why is the y vs. t graph still a flat line? That's because the y-position of the ball is constant. It never changes, so the y-velocity is zero and remains zero. Note that y is not zero. It's vy that is zero.
The mathematical equation that describes the relationship between x and t on a graph of constant acceleration is
where x0 is the x-position at t=0 and v_x0 is the x-velocity at t=0.
Other useful equations include
1. Suppose that a motorcycle can accelerate from 0 to 30 m/s in 2.5 s with constant acceleration. What is its acceleration? (Assume it is moving in the +x direction)
2. Suppose you fall from a height of 3 m (starting from rest) and have a constant acceleration of 10 m/s2 downward. How fast will you be moving when you hit the ground? Note: it's easiest to define the +y direction as downward. However, you don't have to. You can define the +y direction as upward. Just remember to set ay =-10 m/s2, and when you solve for vy and get a negative answer, remember that the question is asking for speed.
3. A golf ball rolls as shown in the animation below.
(a) What are the components of the acceleration of the golf ball?
ax = <_> m/s2 ay = <_> m/s2
(b) What are the x and y components of the displacement of the golf ball between t = 1 s and t = 3 s?
We've looked at a position vs. time graph for motion with constant acceleration and noticed that it is curved, showing that the velocity component is changing. What about the velocity vs. time graph for motion with constant acceleration? Let's look back at the golf ball that moves linearly in the +x direction with an acceleration also in the +x direction.
Notice that the graph of vx vs. t is a line of constant slope. The slope on a vx vs. t graph is the x-acceleration. Likewise, the slope on a vy vs. t graph is the y-acceleration, which you will notice is zero in this example.
1. What are the components of the acceleration of the golf ball in the previous animation?
ax = <_> m/s2 ay = <_> m/s2