https://file.aiquickdraw.com/mathbot/file/1725874908863ydp01dyr.jpg

Let's go through the questions one by one from the image. 

### 1) **Kinematics in 1D**
The first problem involves a car's motion along a straight line, given by the position function:
\[
x(t) = 5t^2 + 2t + 1
\]
where \(x\) is in meters and \(t\) is in seconds.

- **(a)** **Position at \(t = 3\):**
Substitute \(t = 3\) into the position function to find the car's position at that time.

- **(b)** **Velocity at \(t = 3\):**
The velocity is the first derivative of the position function, \(v(t) = \frac{dx(t)}{dt}\). Compute the derivative of \(x(t)\) and then substitute \(t = 3\).

- **(c)** **Acceleration:**
The acceleration is the second derivative of the position function, \(a(t) = \frac{d^2x(t)}{dt^2}\). 

### 2) **Analyzing velocity and acceleration from a graph**
The second problem gives a velocity vs. time graph with two points: \(Q\) and \(P\). You're asked to determine:

- **(a)** **Acceleration of \(Q\):**
Acceleration is the slope of the velocity-time graph. Calculate the slope between the origin and \(Q\).

- **(b)** **Acceleration of \(P\):**
Find the slope of the line from the origin to \(P\).

- **(c)** **Time when \(P\) and \(Q\) meet:**
You need to determine when both lines have the same velocity.

- **(d)** **Distance from the initial position when \(P\) and \(Q\) meet:**
This can be found using the area under the velocity-time graph.

- **(e)** **Velocity when \(P\) and \(Q\) meet:**
This is the velocity where the two lines intersect.

### 3) **Projectile motion**
The third problem is a projectile motion question, where a bullet is fired at a target 91.4 meters away with an initial velocity \(v_0 = 427 \, \text{m/s}\).

- **(a)** **Find two possible angles (\(\theta_1\) and \(\theta_2\)):**
Use the given identity for projectile motion and solve for the angles that satisfy the range equation.

- **(b)** **Time of flight:**
Determine the time required for the bullet to reach the target for both angles.

Would you like me to provide detailed solutions for any specific part of this problem set?

Physics Problems related to Kinematics in 1D and 2D: Analyze car motion, velocity vs time graph, and projectile motion.

Solve kinematics problems in 1D and 2D. Analyze car motion using position, velocity, and acceleration formulas, interpret a velocity-time graph, and calculate angles for projectile motion. Suitable for first-year physics students.

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