Vectors & Motion (College Board AP® Physics 1: Algebra-Based): Free Response Questions

Figure 1

Two identical spheres are released from a device at time $t=0$ from the same height $H$, as shown above. The device is shown in Figure 1.

Sphere A has no initial velocity and falls straight down. Sphere B is given an initial horizontal velocity of magnitude $v_0$ and travels a horizontal distance $D$ before it reaches the ground. The spheres reach the ground at the same time $t_f$, even though Sphere B has more distance to cover before landing. Air resistance is negligible.

The dots in Figure 2 represent Spheres A and B. Draw a free-body diagram showing and labeling the forces (not components) exerted on each sphere at time $\frac{t_f}{2}$ .

Figure 2

Derive an expression for the total time of flight of each sphere, in terms of $H$ and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference book.

On the axes in Figure 3, sketch and label a graph of the horizontal component of the velocity of Sphere A and of Sphere B as a function of time.

Figure 3

Justify whether your answers to part b) and part c) support the finding that the spheres reach the ground at the same time, even though they travel different distances.

Figure 1

A clay ball of mass $m$ is launched at an angle $\varphi$ above the horizontal with initial speed $v_0$. At the moment it reaches the highest point in its trajectory and is moving horizontally, it collides with and sticks to a wooden block of mass $M$, as shown in Figure 1. The block is suspended from a light string of length $L$. The block and the clay then swing up to a maximum height $h_0$ above the block’s initial position and make an angle $\theta$ with the vertical.

On the axes provided in Figure 2, sketch and label graphs of the horizontal and vertical components of the velocity of the clay ball as a function of time from the time the ball is launched to the time it strikes the block.

Figure 2

A projectile is fired from level ground with speed $v_0$ at an angle $\theta >45°$ to the ground. The projectile is fired from a few centimeters before position $x_1$, reaches its maximum height at position $x_2$, and lands on the ground at position $x_{3.}$.

i) Derive an expression for the maximum height of the projectile in terms of $v_0$, $g$, $\theta$ and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference book.

ii) Determine an expression for the mass of the projectile in terms of maximum gravitational potential energy $U_{gmax}$, $v_0$, $\theta$ and physical constants as appropriate.

Figure 1

A stunt cyclist builds a ramp that will allow the cyclist to coast down the ramp and jump over several parked cars, as shown in Figure 1. To test the ramp, the cyclist starts from rest at the top of the ramp, then leaves the ramp, jumps over six cars, and lands on a second ramp. The launch point and the landing point are at the same height. Friction and air resistance are negligible.

Figure 1 shows the vertical distance between the top of the first ramp and the launch point, $H_0$, and the angle of the ramp at the launch point, ${\theta }_0$, as measured from the horizontal. The cyclist travels a horizontal distance of $X_0$ while the cyclist and bicycle are in the air. The combined mass of the stunt cyclist and bicycle is $m_0$.

i) Starting with conservation of energy, determine an expression for the speed of the cyclist as they leave the ramp in terms of $H_0$ and physical constants, as appropriate.

ii) Starting with a kinematic equation, derive an expression for the horizontal distance $X_0$ in terms of $H_0$, ${\theta }_0$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

In experiment 2, the ramp is modified so that the cyclist now starts from a height of $2H_0$.

Indicate whether the horizontal distance covered by the stunt cyclist in experiment 2, $X_{new}$, will be less than, equal to, or greater than $X_0$ by writing one of the following:

• $X_{new}<X_0$

• $X_{new}=X_0$

• $X_{new}>X_0$

Justify your reasoning. In your justification, include qualitative reasoning beyond mathematical derivations or expressions.