Matter.js for collision detection

Solution 1:

You would have to redo most of your code, but the movement code CAN be moved over fairly easily.

First you need to make a new engine and run it:

//Create engine - All the game stuff
var Engine = Matter.Engine,
    Render = Matter.Render,
    Runner = Matter.Runner,
    Composites = Matter.Composites,
    Common = Matter.Common,
    World = Matter.World,
    Bodies = Matter.Bodies,
    Body = Matter.Body;

// create an engine
var engine = Engine.create(),
    world = engine.world;

// create a renderer
var render = Render.create({
    canvas: document.getElementById("canv"),
    engine: engine,
    options: {
        width: 500,
        height: 500,
        wireframes: false,
        background: '#6DDA4A'
    }
});
engine.world.gravity.y = 0;
Render.run(render);
// create runner
var runner = Runner.create();
Runner.run(runner, engine);

Next, you should make and add a new car object and the block into the world. You can edit these values to whatever you need them to be. Note: rot is not an actual parameter you need, I'm just using it to set the rotation of the car later.

var car = Bodies.rectangle(100, 100, 50, 80, {
    friction: 1,
    frictionAir: 0.1,
    rot: 0,
    restitution: 0,//Makes it so the car won't bounce off of objects
    render: {
        fillStyle: "#FF0000",
        /*sprite: {
            //You can use this to apply a background image to the car
            texture: "Path/To/Image.png",
            xScale: number,
            yScale: number
        }/**/
    }
});
var block = Bodies.rectangle(350, 100, 100, 400, {
    isStatic: true,//Makes block unmovable
    friction: 1,
    frictionAir: 0.1,
    rot: 0,
    restitution: 0,
    render: {
        fillStyle: "#0000FF",
        /*sprite: {
            texture: "Path/To/Image.png",
            xScale: number,
            yScale: number
        }/**/
    }
});
World.add(world, [car, block]);

For updating the car, you can either use body.setPosition or body.applyForce. Since it's a car, I would use body.applyForce so that the car keeps rolling after you've stopped pressing a button. Rotation can also be done with body.setAngle or body.rotate. This time, I'm going to use body.setAngle so the turning feels better.

function updateCar() {
    //Declare variables for velocity

    var speed = 5;
    var carRot = car.rot*180/Math.PI;
    var velY = speed * Math.cos(carRot * Math.PI / 180);
    var velX = speed * Math.sin(carRot * Math.PI / 180)*-1; 
    var pushRot = 0;
    //Update variables

    if (upPressed==false&&downPressed==false) {
        velY = 0;
        velX = 0;
    }
    else {
        //alert(carX+", "+carY);
    }
    if (downPressed == true) {
        velY *= -1;
        velX *= -1;
    }
    if (leftPressed) {
        pushRot = -0.1;
    }
    if (rightPressed) {
        pushRot = 0.1;
    }
    car.rot += pushRot;


    //Set position of car
    carX += velX;
    carY += velY;
    Body.applyForce(car, {x:carX,y:carY}, {x:velX/200,y:velY/200});
    Body.setAngle(car, car.rot);
    requestAnimationFrame(updateCar);
}
window.requestAnimationFrame(updateCar);

Check out a demo here