A ball thrown vertically upward is released at time 𝑡 equals zero. If air resistance is negligible, which of the following graphs best represents the velocity 𝑣 of the ball as a function of time, 𝑡?
The question gives us five velocity-versus-time graphs to choose from, each with the same set of coordinate axes. The vertical axis is 𝑣, velocity. And the horizontal axis is 𝑡, time. And the two axes meet at the origin, where velocity and time are both zero. The question asks us which of these graphs best represents the velocity of a ball thrown vertically upward as a function of time if the ball was released at time 𝑡 equals zero.
Let’s first figure out how the ball’s velocity should change with time and then figure out which graph matches our expectations. Here’s our ball. And at time 𝑡 equals zero, it’s thrown vertically upward. Since the ball is thrown, it has some initial velocity. Let’s call this 𝑣 zero for the velocity at time 𝑡 equals zero. And of course, since the ball is thrown upward, 𝑣 zero points up. Already, we can eliminate choice D since the initial velocity in choice D is zero, which is inconsistent with the ball thrown upward and released at time 𝑡 equals zero with some nonzero initial velocity. Choices A, B, C, and E are consistent with what we know so far since they all have a nonzero initial velocity. Note that we do still have to keep considering choice B with a negative initial velocity, since the question doesn’t specify if negative velocity points upwards or downwards.
The next step to working out what we expect the answer to look like is to figure out what factors contribute to a changing velocity. One factor that could change the velocity — specifically slow it down — is air resistance. But we’re told in the problem statement that the air resistance is negligible. So, it’s not something that we need to consider. Any other acceleration would also tend to change the velocity of the ball, since acceleration is by definition a change of velocity, in this case, with the presence of a gravitational field. So, there is an acceleration pulling downward on the ball. Since there is an acceleration acting on the ball, the velocity must change with time. We can, therefore, eliminate choice C since in choice C, the velocity is constant. It doesn’t change with time and it has zero acceleration.
This leaves us with choices A, B, and E, all of which have the velocity changing with time and so have nonzero acceleration. Let’s try to distinguish between A, B, and E by comparing the relative direction of velocity and acceleration. Recall that a negative acceleration tends to make velocity more negative, while a positive acceleration tends to make velocity more positive.
We can see then that choice A has a negative acceleration since its velocity is decreasing with time. Choice B’s velocity is also decreasing with time. So, choice B also has a negative acceleration. Choice E has a positive acceleration since its velocity is increasing with time. In our problem, the acceleration is the gravitational acceleration, which we’ll call 𝑔. And 𝑔 points down. We still don’t know if down corresponds to the positive or negative direction. But we can now compare the relative directions of velocity and acceleration.
The initial velocity 𝑣 zero points up and the acceleration 𝑔 points down. So, we need a graph whose acceleration and initial velocity point in opposite directions. Choice E does not satisfy this condition since the initial velocity is positive and the acceleration is also positive. Choice B does not satisfy this condition either, since the initial velocity is negative and the acceleration is also negative. However, choice A does satisfy this condition, since the initial velocity is positive, but the acceleration is negative, which means the velocity and the acceleration point in opposite directions, exactly what we’re looking for.
Let’s summarize our expectations for the correct velocity-versus-time graph should look like. First, since the ball is thrown upward and released at 𝑡 equals zero, the initial velocity is not zero. Second, there is nonzero acceleration, in this case due to gravity. Our third and last expectation is that the initial velocity and the acceleration point in opposite directions, which means they have opposite signs. Observation one allowed us to eliminate choice D. Observation two allowed us to eliminate choice C. Observation three allowed us to eliminate choices B and E. And choice A is the one that meets all three of our expectations. So, choice A is the correct answer.
We actually could have solved this problem with only one expectation no matter how velocity changes with respect to time. We’ll consider the trajectory of the ball as it’s thrown. Initially, the ball is thrown upward. Then, it travels up. And eventually, gravity pulls it back down. Remember that the velocity points in whatever direction the ball travels. So, when the ball is travelling up, the velocity points up. And when the ball travels down later on, the velocity points down. This means that the correct graph of velocity versus time will have the velocity changing direction at some point, which is the same thing as saying that the velocity will change sign.
Of the answer choices, B, C, D, and E don’t have a velocity that changes sign. However, answer choice A has a velocity that’s initially positive and later on is negative. So, only answer choice A meets the expectation that at some time the velocity changes sign.