If the applied force is constant, what is the graph of the velocity?

Question

This is a sample interview question from Oxford:

A ball, initially at rest, is pushed upwards by a constant force for a certain amount of time. Sketch the velocity of the ball as a function of time, from start to when it hits the ground.

My understanding is that the force is continuously applied since the force is "constant" (do correct me if I'm wrong), and then I tried to visualize the position function but I just got confused. I couldn't picture if it's linear or exponential.

So instead, I tried to solve for $v$ using $F = ma$.

$$ F = ma\\ F = m\frac{dv}{dt}\\ \frac{dv}{dt} = \frac{F}{m}\\ \int\frac{dv}{dt}dt = \frac{F}{m}\int dt\\ v = \frac{F}{m}t $$

which means the velocity graph is linear.

• Am I doing things correctly?
• What are the other ways of solving/thinking about this especially non-mathematical ones?

Thank you.

Answer 1

As the force $F$ is constant, and we know that the force of weight $mg$ is constant too; we can say the net force acting on the ball is constant whole the motion. But, from $t=0$ to $t=t_1$ the net force is $F-mg$ and after $t=t_1$ the net force is $-mg$. So, velocity graph will be linear but it has a breaking at $t=t_1$. For $0\le t\lt t_1$ the gradient of the graph is $\frac{F-mg}m$ and for $t\gt t_1$ the gradient of the graph is $-g$.

Answer 2

I agree with your deduction. The speed is linearly growing, as you have proven.

You can arrive at this conclusion in a quicker way by considering Newton's 2nd law. A constant force will cause a constant acceleration, which in turn causes a constant increase per time unit in the speed. So, the speed will grow linearly.